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Statistical Analysis/Unit 4 Navigation

This is the main navigation page for Unit 4 of the course Introduction to Statistical Analysis, developed using openly licensed materials from Saylor.org's Introduction to Statistics. Below you will find a full description of Unit 4 in general, as well as for each subunit. Follow the links within each subunit description to access particular topics, or proceed directly to the Unit 4 Content Page.

UNIT 4: CENTRAL LIMIT THEOREM AND CONFIDENCE INTERVALS
In this unit, you will learn how to use the central limit theorem and confidence intervals, the latter of which enables us to estimate unknown population parameters. The central limit theorem provides us with a way to make inference from samples of non-normal populations. This theorem states that given any population (and it does not matter whether or not it is a normal distribution), as the sample size increases, the sampling distribution of the means approaches a normal distribution. It is a powerful theorem because it allows us to assume that given a large enough sample, the sampling distribution will be normally distributed. The central limit theorem is one of the most important ideas in statistics, so be sure to spend time on it. In this unit, you will also learn about confidence intervals, which provide us with a way to estimate a population parameter. Instead of giving just a one-number estimate of a variable, a confidence interval gives a range of likely values for it. This is useful because sample results will vary from sample to sample, so a range of values is better than a one-number estimate. After completing this unit, you will know how to construct confidence intervals and calculate the margin of error for them. You will learn to how to come up with a range of values for a parameter and the level of confidence for the intervals.

For example, suppose you want to know the amount of soda that an average high school student in New York drinks per day. The average volume of soda for the entire population of New York high school students who drink soda is the parameter you are trying to estimate. Suppose you take a random sample and find out the average amount is 0.5 litres. Then you also want to know how much you expect the average to vary from one sample to the next, with a certain level of confidence. The number that you use to represent this precision, i.e. to measure how close you expect your results to be to the truth, is called the margin of error.

Time Advisory
Time Advisory: This unit will take you 11 hours to complete.
 * Subunit 4.1: 5 hours
 * Subunit 4.2: 6 hours

Learning Outcomes
Upon completion of this this unit, you will be able to:
 * Recognize the Central Limit Theorem problems.
 * Classify continuous word problems by their distributions.
 * Apply and interpret the Central Limit Theorem for Averages.
 * Apply and interpret the Central Limit Theorem for Sums.
 * Calculate and interpret confidence intervals for one population average and one population proportion.
 * Interpret the student-t probability distribution as the sample size changes.
 * Discriminate between problems applying the normal and the student-t distributions.

Subunits
Unit four consists of two main topics:

The Central Limit Theorem

 * 4.1.1: The Central Limit Theorem for Sample Means (Averages)
 * 4.1.2: The Central Limit Theorem for Sums
 * 4.1.3: Using the Central Limit Theorem

Confidence Intervals

 * 4.2.1: Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal
 * 4.2.1: Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T
 * 4.2.1: Confidence Interval for a Population Proportion

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 4 Navigation


text:
Statistical Analysis/Unit 4 Content

Unit 4
This content is adapted from the Introduction to Statistics MA121/ECON104 Course at Saylor.org.

The Central Limit Theorem for Sample Means (Averages)

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem”
 * “Section 1: The Central Limit Theorem”
 * “Section 2: The Central Limit Theorem for Sample Means (Averages)”

Instructions: Please read each of the linked sections above in their entirety.

 * Lecture: Khan Academy’s Statistics
 * “Central Limit Theorem”

Instructions: Please view the lecture in its entirety (approximately 10 minutes). This lecture will provide and introduction to the central limit theorem and the sampling distribution of the mean.

 * Lecture: Khan Academy’s Statistics
 * “Sampling Distribution of the Sample Mean”
 * “Sampling Distribution of the Sample Mean 2”

Instructions: Please view the both lectures, which discuss the sampling distribution of the sample mean, in their entirety (approximately 11 minutes and 13 minutes, respectively).

 * Lecture: Khan Academy’s Statistics
 * “Standard Error of the Mean”
 * “Sampling Distribution Example Problem

Instructions: Please view both lectures in their entirety (approximately 30 minutes total). These lectures will discuss the standard error of the mean, i.e. the standard deviation of the sampling distribution of the sample mean, and work out an example.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

The Central Limit Theorem for Sums

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem"
 * "Section 3: The Central Limit Theorem for Sums”

Instructions: Please read the entire webpage in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Using the Central Limit Theorem

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem"
 * "Section 4: Using the Central Limit Theorem”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video lecture 7: The Central Limit Theorem”

Instructions: Please view the lecture to the right in its entirety.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 7: The Central Limit Theorem”
 * “Practice: The Central Limit Theorem”

Instructions: Click on the hyperlink titled “Practice: The Central Limit Theorem”. Please solve all the problems in this section. Next, click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19-25. The solutions are provided below the problem. Please solve all of the problems before checking the solutions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals”
 * “Section 1: Confidence Intervals”
 * “Section 2: Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal”

Instructions: Please read each of the linked sections above in their entirety.

 * Lecture: Khan Academy’s Statistics
 * “Confidence Interval 1”

Instructions: Please view the lecture in its entirety (approximately 14 minutes). In this lecture, you will learn to estimate the probability that the true population mean lies within a given range around a sample mean.

 * Lecture: Khan Academy’s Statistics
 * “Margin of Error 1”
 * “Margin of Error 2”

Instructions: Please view both lectures in their entirety (approximately 25 minutes total). In these lectures, you will learn to find the 95% confidence interval for a problem.

 * Lecture: Khan Academy’s Statistics
 * “Confidence Interval Example”

Instructions: Please view the lecture linked above (approximately 19 minutes). In this lecture, you will work out a confidence interval example

Terms of Use: Please respect the copyright and terms of use displayed on the webpage displayed above.

Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals"
 * "Section 3: Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Khan Academy’s Statistics
 * “Small Sample Size Confidence Intervals”
 * “Small Sample Hypothesis Test”

Instructions: Please view both lectures in their entirety (approximately 21 minutes total). These lectures will discuss constructing small size confidence intervals using t-distributions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages displayed above.

Confidence Interval for a Population Proportion

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals"
 * "Section 4: Confidence Interval for a Population Proportion”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video lecture 8: Confidence Intervals”

Instructions: Please view the lecture to the right in its entirety.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 8: Confidence Intervals”
 * “Practice 1: Confidence Intervals for Averages, Known Population Standard Deviation”
 * “Practice 2: Confidence Intervals for Averages, Unknown Population Standard Deviation”
 * “Practice 3: Confidence Intervals for Proportions”

Instructions: Click on the hyperlink titled “Practice 1: Confidence Intervals for Averages, Known Population Standard Deviation” and solve problems 1-13. Next, click on the hyperlinks titled “Practice 2: Confidence Intervals for Averages, Unknown Population Standard Deviation” to solve problems 1-11 and “Practice 3: Confidence Intervals for Proportions” to solve problems 1-13. Finally, click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. The solutions are provided below the problem. Please solve all of the problems before checking the solutions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Statistical Analysis/Unit 4 Content


text:
User:EGM6321.f09.team1.Zhichao Gong/Mtg2

Mtg 2: Thu, 27 Aug 09

page2-1

 * PEA1.F09.Mtg2.pg1.fig1.svg

p. 1-2, Eqs (1), (2)

f(s,t)

$$S \equiv {X}^{1}= \ axial \ coordinate$$

$$t \ = \ time$$

$$ \begin{cases} & X \ = \ capital \ (uppercase) \\ & x \ = \ small \ (lowercase) \end{cases}$$

$$ \begin{cases} & Y \ = \ uppercase \\ & y \ = \ lowercase \end{cases}$$ $$At \ S={Y}^{1}(t), \ have f({Y}^{1},t) \ and \ time \ t$$ page2-2

HW: Derive Eqs (1), (2) p.1-2 Montesquieu

p.1-2Eq(3)       Eq. of motion of wheel(vehicle)

Note: $$\overset{.}{Y}^{1}$$ Math struct of ci(Y1,t) known

but values of ci not known at any t

terms not due to damping,

but due to chain rulel called

"convective terms"

Similarly for

$${c}_{i}({Y}^{1},t), \ i=1,2,3$$

Note: Math struct of ci(Y1,t) known

but values of ci not known at any t

page2-3

Since \ $${\color{red} \left\{ {\color{black}{Y}^{1} \ {u}^{1} \ {u}^{2}}\right\}}$$

are unknown functions to be solved for.

There are 2 more equations of motion for

u1(s,t) and u2(s,t)
 $${\color{red} \Rightarrow \ PDEs}$$

Note: Result is a set of coupled,

nonlinear ODE and PDEs

Simplify:

2nd order to 1st order

nonlinear to linear

unknown varying coefficient to known varying coefficient


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Learn Kannada

I found that there is a lot of enthusiasm in my colleagues,friends and neighbors for learning Kannada. I thought there would be help available in Wikiversity but could not find one such page. It prompted me to start one such page. Here we are! Welcome to this page and to many more lessons to follow.I am Jaikumar Mariappa from the field of translation(Kannada-English).

Active Voice                             Simple Present tense
Word: Take(verb) =Tegeduko = ತೆಗೆದುಕೋ तेगेदुको । 1. I take. Naanu tegedu koɫɫu ttene ನಾನು ತೆಗೆದುಕೊಳ್ಳುತ್ತೇನೆ. नानु तेगेदुकोल्लुत्तेने ।

2. I donot take. Naanu tegedu koɫɫuvu dilla  ನಾನು ತೆಗೆದುಕೊಳ್ಳುವುದಿಲ್ಲ.नानु तेगेदुकोल्लुवुदिल्ल ।

3. We take. Naavu tegedu koɫɫu tteve ನಾವು ತೆಗೆದುಕೊಳ್ಳುತ್ತೇವೆ   नावु तेगेदुकोल्लुत्तेवे ।

4. We donot take. Naavu tegedu koɫɫu vudilla  ನಾವು ತೆಗೆದು ಕೊಳ್ಳುವುದಿಲ್ಲ. नावु तेगेदुकोल्लुवुदिल्ल ।

5. You take. Neevu tegedu koɫɫi(respectful) (plural)        ನೀವು ತೆಗೆದುಕೊಳ್ಳಿ. नीवु  तेगेदुकोल्ली ।

neenu tegedu ko(singular) ನೀನು ತೆಗೆದುಕೊ. नीनु तेगेदुको ।

6. Do you take ? Neevu tegedu koɫɫu ttiiraa? ನೀವು ತೆಗೆದು ಕೊಳ್ಳುತ್ತೀರಾ? नीवु  तेगेदुकोल्लुत्तीरा ?

7. You donot take. Neevu tegedu koɫɫa bɛɛd̪i ನೀವು ತೆಗೆದು ಕೊಳ್ಳಬೇಡಿ.नीवु   तेगेदुकोल्लबेड़ी।

8. Donot you take ? Neevu tegedu koɫɫu vudilla vaa? ನೀವು ತೆಗೆದು ಕೊಳ್ಳುವುದಿಲ್ಲವಾ? नीवु  तेगेदुकोल्लुवुदिल्लवा?

9. They take. avaru tegedu koɫɫu ttaare ಅವರು ತೆಗೆದು ಕೊಳ್ಳುತ್ತಾರೆ. अवरु तेगेदुकोल्लुत्तारे ।

10. Do they take ? avaru tegedu koɫɫu ttaa raa? ಅವರು ತೆಗೆದು ಕೊಳ್ಳುತ್ತಾರಾ? अवरु तेगेदुकोल्लुत्तारा?

11. They donಕಿt take. avaru tegedu koɫɫu vudilla. ಅವರು ತೆಗೆದು ಕೊಳ್ಳು ವುದಿಲ್ಲ. अवरुतेगेदुकोल्लुवुदिल्ल।

12. Donಕಿt they take ? avaru tegedu koɫɫu vudilla vaa? ಅವರು ತೆಗೆದು ಕೊಳ್ಳು ವುದಿಲ್ಲವಾ? अवरुतेगेदुको ल्लुवुदिल्लवा ?

Note: for ɫ,d̪, please touch the tongue to the palate.यहा ल्ल उच्चारण करनेका वक्त जिह्वा तालू(palate) को ताड़ना चाहिए । ɛ is pronounced as 'a' in mare 9,10,11,12 can be used in case of he,she with respect Try  ittuko इट्टुको=keep,ettiko= एततिको lift, kEElisiko केलिसिको=listen  substituting them for tegeduko तेगेदुको ।

--

This and That(simple sentences)
Unlike the Indo -European Languages (English or Hindi,) in Kannada (probably in all Dravidian languages) we can naturally construct proper simple sentences without using words indicating the verb and articles. Well, I will explain,

This is a book. is a simple sentence in English. (यह एक पुस्तक है|)  In Kannada simple sentences can be constructed without the words in Italics. This is a book. (यह एक पुस्तक  है| )

This = ಇದು idu (I is pronounced as in IF ) इदु ,

That = ಅದು = Adu अदु

ಇದು ಮನೆ.(Idu mane) इदु मने

ಇದು ಪುಸ್ತಕ. (pusthaka)इदु पुस्तक

ಅದು ಬೆಟ್ಟ(betta)mountain बेट्ट and ಇದು ತೋಟ thOOta(garden)तोट

हिन्दी लिप्यंतर मे अंतिम अ को पूरा उच्चारण करना चाहिए|

ಇದು ಅಂಗಡಿ angadi अंगड़ी

ಅದು ನವಿಲು navilu नविलु

ಇದುನಾಯಿಮರಿ नाईमरी naayimari

ಅದು ಆನೆ आने Aane

ಇದು ಕಾಡು  kaadu काडु,........ಇದು ಬಂಡೆ  bande  बंडे,.........ಇದು ರಸ್ತೆ  raste रस्ते

ಇದು ಗುಡಿ गुड़ी gudi

ಅದು ಚಿಟ್ಟೆ, चिट्टे,chitte

key words: ಇದು, ಅದು , ಮನೆ, ನಮ್ಮ, ನಿಮ್ಮ, ಬೆಟ್ಟ, ತೋಟ, ಅಂಗಡಿ, ನವಿಲು, ನಾಯಿಮರಿ, ಆನೆ, ಕಾಡು, ಬಂಡೆ, ರಸ್ತೆ ,ಗುಡಿ,ಚಿಟ್ಟೆ,

Lesson-3
conversation -1 Conversation makes language learning very interesting and gives life to it.For beginners Bangalore Kannada with good sprinkle of English (Kanglish) will be really helpful. ಸ್ವಲ್ಪ adjust ಮಾಡಿ. (Little bit adjust) ಇದು ಮನೆ. (This house)

title: Learn Kannada


text:
Template:Critique of article

(Review Paper) Cited in

title: Template:Critique of article


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User talk:Jm7623

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title: User talk:Jm7623


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Payroll

Payroll is one of a series of accounting transactions dealing with the process of paying employees for services rendered, after processing of the various requirements for withholding of money from the employee for payment of payroll taxes, insurance premiums, employee benefits, garnishments and other deductions.

About Payroll
Payroll involves the calculation of amounts due the employee, such as hourly wages, a salary consisting of a certain amount per calendar period, or pay to salespersons on commission, as well as reimbursement for employee-paid expenses such as travel (calculated either based on actual amount paid by the employee or utilizing a per diem rate).

In addition to these payments, there are often computed amounts of paid vacation time and accrued sick leave which have been used or are available for use. These are simply carried as bookkeeping entries as available or used and accumulate until the employee actually uses them. If an employee collects vacation pay or sick leave, these amounts are then added to the amount due the employee to be paid.

All of these monies credited to the employee are usually referred to as gross pay.

From these amounts that are credited to the employee, various debits are taken as withholding, the most significant being income tax, then other taxes such as social security and Medicare (in the United States; other countries have similar programs for the collection of government-mandated retirement benefit supplement systems). There may also be additional deductions for supplemental health insurance, union dues, pension plan contributions, garnishment for nonpayment of debts, repayment of prior salary, vacation or sick leave overpayments made in error, undercollection of insurance, and other deductions.

The amount left after deductions from gross pay is generally the amount given in the employee's pay envelope, either as cash or a check. This amount is known as net pay. If the employee has their net pay deposited to their bank account (through a process known as direct deposit) then the employee may simply receive a pay stub indicating this.

In addition to amounts collected from the employee, frequently the employer is required to also pay additional monies of their own such as government taxes such as employer matching contribution to Social Security, and employer-paid systems such as unemployment insurance; employer matching for pension plans such as 401(k); employer portion (or all) of employee health insurance, and additional expenses.

Organizations which have very few employees may perform payroll by hand. Business organizations which have become too large to perform such tasks by hand (or small ones that prefer not to do them by hand) will generally use an accounts software, or an integrated module from a Human Resource Management System on a computer to perform this task, or may pay a service bureau to perform the process of payroll calculation and writing of payroll checks.

Due to government mandated (and often severe) penalties for improper or inadequate collection of payroll taxes and paying of wages, almost all employers (other than very tiny ones having perhaps one or very few employees) either use a payroll computer program or a service bureau.

Payroll taxes in U.S.
Before considering the payroll taxes we need to talk about the Basic Formula for the Net Pay. Basically from gross pay is subtracted one or more deductions to arrive at the Net Pay. In fact Employee's gross pay (pay rate times number of hours worked, including any over time) minus payroll tax deductions, minus voluntary payroll deductions, is equal to Net Pay. As you can see payroll tax deductions play a critical role and just because they are provided by law we can call them Statutory payroll tax deductions.

The employer must withhold payroll taxes from an employee's check and hand them over to several tax agencies by law. Payroll taxes include:
 * 1) Federal income tax withholding, based on withholding tables in "Publication 15, Employer's Tax Guide" by Internal Revenue Service - IRS;
 * 2) Social Security tax withholding. The employee pays 6.2 percent of the salary or wage, up to 106,800. The employer also pays 6.2 percent in Social Security taxes. If you are self-employed, you pay the combined employee and employer amount of 12.4 percent in Social Security taxes on your net earnings;
 * 3) Medicare tax. The employee pays 1.45 percent in Medicare taxes on the entire salary or wage. The employer also pays 1.45 percent in Medicare taxes. If you are self-employed, you pay the combined employee and employer amount of 2.9 percent in Medicare taxes on your net earnings;
 * 4) State income tax withholding;
 * 5) various local tax withholding, such as city taxes, county taxes, school taxes, state disability, and unemployment insurance.

As for the sources considered as references we can mention the following publications:
 * Publication 15, (Circular E), Employer's Tax Guide. This publication explains employer's tax responsibilities. It explains the requirements for withholding, depositing, reporting, paying, and correcting employment taxes. It explains the forms any employer must give to its employees, those employees must give to the employer, and those employer must send to the IRS and SSA (Social Security Administration). This guide also has tax tables needed to figure the taxes to withhold from each employee;
 * Publication 15 - A, Employer’s Supplemental Tax Guide. This publication supplements Publication 15 (Circular E), Employer’s Tax Guide. It contains specialized and detailed employment tax information supplementing the basic information provided in Publication 15 (Circular E);
 * Publication 15-B. Employer's Tax Guide to Fringe Benefits. This publication supplements Publication 15 (Circular E), Employer’s Tax Guide, and Publication 15 - A, Employer’s Supplemental Tax Guide. This publication contains information about the employment tax treatment of various types of noncash compensation.

In the earlier part we have considered payroll taxes related to employee's side. Now it's the moment to talk about the Employer Payroll Taxes Employers are responsible for paying their portion of payroll taxes. These payroll taxes are an expense over and above the expense of an employee's gross pay. The employer-portion of payroll taxes include the following:
 * 1) Social Security taxes (6.2% up to the annual maximum);
 * 2) Medicare taxes (1.45% of wages);
 * 3) Federal unemployment taxes (FUTA);
 * 4) State unemployment taxes (SUTA).

Very often you can hear people using FICA in their terminology. FICA stands for the Federal Insurance Contributions Act and the FICA tax consists of both Social Security and Medicare taxes. As we explained earlier both parties pay half of these taxes. Employees pay half, and employers pay the other half. Social Security and Medicare taxes are paid both by the employees and the employers. In summary together both halves of the FICA taxes add up to 15.3 percent.

Any employer is responsible for paying the employer's share of payroll taxes, for depositing tax withheld from the employees' paychecks, preparing various reconciliation reports, accounting for the payroll expense through their financial reporting, and filing payroll tax returns. As you see this suite of employer payroll tax responsibilities is far above issuing paychecks to employees.

Different Types of Accounting

 * Financial transaction
 * Accounts payable
 * Accounts receivable
 * Trial balance

Other Sources

 * Institute of Payroll Professionals official website
 * Glossary of Accounting Terms
 * Independent research, seminars and consulting on multi-country payroll
 * Direct Deposit information resource
 * Publication 15 (2010)(Circular E) Employer's Tax Guide

title: Payroll


text:
Statistical Analysis/Unit 5 Navigation

This is the main navigation page for Unit 5 of the course Introduction to Statistical Analysis, developed using openly licensed materials from Saylor.org's Introduction to Statistics. Below you will find a full description of Unit 5 in general, as well as for each subunit. Follow the links within each subunit description to access particular topics, or proceed directly to the Unit 5 Content Page.

UNIT 5: HYPOTHESIS TESTING
One of the major goals in statistics is to use the information you collect from a sample to get a better idea of the entire population you’re interested in. In this unit, you will learn about hypothesis testing, which enables us to achieve that goal.

A hypothesis test involves collecting and evaluating data from a sample. The data gathered and evaluated is then used to make a decision as to whether or not the data supports the claim that is made about the population. This unit will also teach you how to conduct hypothesis tests and to identify and differentiate between the errors associated with them.

Many times, you need answers to questions in order to make efficient decisions. For example, a restaurant owner might claim that his restaurant’s food costs 30% less than other restaurants in the area or a phone company might claim that its phones last at least one year more than phones from other companies. You will have to make decisions about these claims. The process of hypothesis testing is a way of decision-making. In this unit, you will learn to establish your assumptions through null and alternative hypotheses. Then, you will learn to compare sample characteristics to assumptions to see whether there is enough data to accept or reject the null hypothesis. The null hypothesis is the hypothesis that is assumed to be true and the hypothesis you hope to nullify, while the alternative hypothesis is the research hypothesis that you claim to be true. This means that you need to conduct the correct tests to be able to accept or reject the null hypothesis. The unit will finally conclude with an introduction to Chi-distributions and their applications.

Time Advisory
Time Advisory: This unit will take you 15 hours to complete.
 * Subunit 5.1: 5 hours
 * Subunit 5.2: 5 hours
 * Subunit 5.3: 5 hours

Learning Outcomes
Upon completion of this this unit, you will be able to:
 * Differentiate between Type I and Type II Errors.
 * Describe hypothesis testing in general and in practice.
 * Conduct and interpret hypothesis tests for a single population mean, population standard deviation known.
 * Conduct and interpret hypothesis tests for a single population mean, population standard deviation unknown.
 * Conduct and interpret hypothesis tests for a single population proportion.
 * Classify hypothesis tests by type.
 * Conduct and interpret hypothesis tests for two population means, population standard deviations known.
 * Conduct and interpret hypothesis tests for two population means, population standard deviations unknown.
 * Conduct and interpret hypothesis tests for two population proportions.
 * Conduct and interpret hypothesis tests for matched or paired samples.
 * Interpret the chi-square probability distribution as the sample size changes.
 * Conduct and interpret chi-square goodness-of-fit hypothesis tests.
 * Conduct and interpret chi-square test of independence hypothesis tests.
 * Conduct and interpret chi-square single variance hypothesis tests.

Subunits
Unit five consists of three main topics:

Hypothesis Testing: Single Mean and Single Proportion

 * 5.1.1: Null and Alternate Hypotheses
 * 5.1.2: Type I and Type II Errors
 * 5.1.3: Distribution for Hypothesis Testing and More

Hypothesis Testing: Two Means, Paired Data, Two Proportions

 * 5.2.1: Comparing Two Independent Population Means with Unknown Population Standard Deviations
 * 5.2.2: Comparing Two Independent Population Means with Known Population Standard Deviations
 * 5.2.3: Comparing Two Independent Population Proportions
 * 5.2.4: Matched or Paired Samples

Chi-Square Distribution

 * 5.3.1: Notation and Facts
 * 5.3.2: Goodness-of-Fit Test
 * 5.3.3: Test of Independence
 * 5.3.4: Test of a Single Variance (Optional)

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 5 Navigation


text:
Statistical Analysis/Unit 5 Content

Unit 5
This content is adapted from the Introduction to Statistics MA121/ECON104 Course at Saylor.org.

Null and Alternate Hypotheses

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 9: Hypothesis Testing: Single Mean and Single Proportion”
 * “Section 1: Hypothesis Testing: Single Mean and Single Proportion”
 * “Section 2: Null and Alternate Hypotheses”

Instructions: Please read each of the linked sections above in their entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Type I and Type II Errors

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 9: Hypothesis Testing: Single Mean and Single Proportion"
 * "Section 3: Outcomes and the Type I and Type II Errors”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Khan Academy’s Statistics
 * “Type 1 Errors”

Instructions: Please watch the brief video lecture (3:24 minutes). This lecture will discuss Type 1 Errors.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage displayed above.

Distribution for Hypothesis Testing and More

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 9: Hypothesis Testing: Single Mean and Single Proportion”
 * “Section 4: Distribution Needed for Hypothesis Testing”
 * “Section 5: Assumption”
 * “Section 6: Rare Events”
 * “Section 7: Using the Sample to Support One of the Hypotheses”
 * “Section 8: Decision and Conclusion”
 * “Section 9: Additional Information”

Instructions: Please read each of the linked sections above in their entirety.

 * Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 9: Hypothesis Testing with a Single Mean”

Instructions: Please view the lecture to the right in its entirety.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 9: Hypothesis Testing: Single Mean and Single Proportion”
 * “Practice 1: Single Mean, Known Population Standard Deviation”
 * “Practice 2: Single Mean, Unknown Population Standard Deviation”
 * “Practice 3: Single Proportion”

Instructions: Click on the hyperlinks titled “Practice 1: Single Mean, Known Population Standard Deviation,” “Practice 2: Single Mean, Unknown Population Standard Deviation,” and “Practice 3: Single Proportion” and solve the problems. Next, click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27. Please click on “[ Show Solution ]” link below the problem to check your solution.

 * Lecture: Khan Academy’s Statistics
 * “Large Sample Proportion Hypothesis Testing”

Instructions: Please view the entire video lecture (14:30 minutes). This lecture will discuss large sample proportion hypothesis testing.

 * Lecture: Khan Academy’s Statistics
 * “Difference of Sample Means Distribution”

Instructions: Please view the entire lecture (approximately 12 minutes). This lecture will discuss the difference of sample means distribution

 * Lecture: Khan Academy’s Statistics
 * “Confidence Interval of Difference of Means”

Instructions: Please watch the entire lecture (approximately 16 minutes). This lecture will discuss confidence interval of difference of means.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage displayed above.

Comparing Two Independent Population Means with Unknown Population Standard Deviations

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions”
 * “Section 1: Hypothesis Testing: Two Population Means and Two Population Proportions”
 * “Section 2: Comparing Two Independent Population Means with Unknown Population Standard Deviations”

Instructions: Please read each of the linked sections above in their entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Comparing Two Independent Population Means with Known Population Standard Deviations

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions”
 * “Section 3: Comparing Two Independent Population Means with Known Population Standard Deviations”

Instructions: Please read the entire webpage in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Comparing Two Independent Population Proportions

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions"
 * "Section 4: Comparing Two Independent Population Proportions”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Khan Academy’s Statistics
 * “Comparing Population Proportions 1”
 * “Comparing Population Proportions 2”
 * “Hypothesis Test Comparing Population Proportions”

Instructions: Please view these three video lectures in their entirety (approximately 27 minutes total). These lectures will discuss comparing population proportions and hypothesis test comparing population proportions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages displayed above.

Matched or Paired Samples

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions
 * "Section 5: Matched or Paired Samples”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 10: Hypothesis Testing with Two Means"

Instructions: Please view the linked lecture.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions”
 * “Practice 1: Hypothesis Testing for Two Proportions”
 * “Practice 2: Hypothesis Testing for Two Averages”

Instructions: Click on the hyperlinks titled “Practice 1: Hypothesis Testing for Two Proportions” and “Practice 2: Hypothesis Testing for Two Averages” and solve the problems. Next, click on the hyperlink titled “Homework” and solve problems 1, 5, 7, 9, 11, 15, 17, 21, 23, 27, 29, 33, 35, 43-48. Please click on “[ Show Solution ]” link below the problem to check your solution.

 * Lecture: Khan Academy’s Statistics
 * “Hypothesis Test for Difference of Means”

Instructions: Please view the entire 10-minute video lecture, which will discuss hypothesis test for difference of means.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage displayed above.

Subunit 5.3: Chi-Square Distribution
5.3.1	Notation and Facts

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 11: The Chi-Square Distribution”
 * “Section 1: The Chi-Square Distribution”
 * “Section 2: Notation”
 * “Section 3: Facts About the Chi-Square Distribution”

Instructions: Please read each of the linked sections above in their entirety.

 * Lecture: Khan Academy’s Statistics
 * “Chi-Square Distribution Introduction”
 * “Pearson's Chi Square Test (Goodness of Fit)”
 * “Contingency Table Chi-Square Test”

Instructions: Please view all three video lectures in their entirety (approximately 40 minutes total). These lectures will provide an introduction to the Chi-Square distribution as well as the Chi-Square test.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages displayed above.

Goodness-of-Fit Test

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 11: The Chi-Square Distribution"
 * "Section 4: Goodness-of-Fit Test”

Instructions: Please read the entire webpage in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Test of Independence

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 11: The Chi-Square Distribution"
 * "Section 5: Test of Independence”

Instructions: Please read the entire webpage in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Test of a Single Variance (Optional)

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 11: The Chi-Square Distribution"
 * "Section 6: Test of a Single Variance (Optional)”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 11: The Chi-Square Distribution”

Instructions: Please view the lecture to the right in its entirety.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 11: The Chi-Square Distribution”
 * “Practice 1: Goodness-of-Fit Test”
 * “Practice 2: Contingency Tables”
 * “Practice 3: Test of a Single Variance”

Instructions: Click on the hyperlinks titled “Practice 1: Goodness-of-Fit Test,” “Practice 2: Contingency Tables,” and “Practice 3: Test of a Single Variance” and solve the problems. Next, click on the hyperlink titled “Homework” and solve odd numbered problems between 3-37. The solutions are provided below the problem. Please solve all of the problems before checking the solutions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 5 Content


text:
Statistical Analysis/Unit 6 Navigation

This is the main navigation page for Unit 6 of the course Introduction to Statistical Analysis, developed using openly licensed materials from Saylor.org's Introduction to Statistics. Below you will find a full description of Unit 6 in general, as well as for each subunit. Follow the links within each subunit description to access particular topics, or proceed directly to the Unit 6 Content Page.

UNIT 6: CORRELATION, REGRESSION, AND ANOVA
One of the main reasons we conduct analysis is in order to understand how two variables are related to one another. The most common type of relationship is a linear relationship. For example, we may want to know what happens to one variable when we increase or decrease the other variable. We want to answer questions such as, “does one increase as the other increases, or does it decrease?” For example, how does drinking soda relate to weight gain for teenagers? Does drinking more soda really relate to more weight gain? In this unit, you will learn to measure the degree of a relationship between two or more variables. Both correlation and regression are measures for comparing variables. However, they are quite different from one another. Correlation quantifies the strength of a relationship between two variables and is a measure of existing data. Regression, on the other hand, is the study of the strength of a linear relationship between an independent and dependent variable, and can be used to predict the value of the dependent variable when the value of the independent variable is unknown. A note of caution: Be careful to not automatically interpret correlation and regression as establishing cause-and-effect relationships!

Finally, you will learn about a method called “Analysis of Variance" (abbreviated ANOVA), which is used for hypothesis tests involving more than two averages. ANOVA is about examining the amount of variability in the y variable and trying to see where that variability is coming from.  You will study the simplest form of ANOVA, called single factor or one-way ANOVA.  Finally, you will briefly study the F distribution, used for ANOVA, and the test of two variances.

Time Advisory
Time Advisory: This unit will take you 11 hours to complete.
 * Subunit 6.1: 2 hours
 * Subunit 6.2: 5 hours
 * Subunit 6.3: 4 hours

Learning Outcomes
Upon completion of this this unit, you will be able to:
 * Discuss basic ideas of linear regression and correlation.
 * Create and interpret a line of best fit.
 * Calculate and interpret the correlation coefficient.
 * Calculate and interpret outliers.
 * Interpret the F probability distribution as the number of groups and the sample size change.
 * Discuss two uses for the F distribution, ANOVA and the test of two variances.
 * Conduct and interpret ANOVA.
 * Conduct and interpret hypothesis tests of two variances (optional).

Subunits
Unit six consists of three main topics:

Linear Regression

 * 6.1.1: Linear Equations, Slope, and Y-intercept
 * 6.1.2: Scatter Plots
 * 6.1.3: The Regression Equation

Correlation

 * 6.2.1: The Correlation Coefficient
 * 6.2.2: Prediction
 * 6.2.3: Outliers

F Distribution and ANOVA

 * 6.3.1: ANOVA
 * 6.3.2: The F Distribution and the F Ratio
 * 6.3.3: The F Distribution and the F Ratio

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 6 Navigation


text:
User:Areil314/sandbox/sample reference

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title: User:Areil314/sandbox/sample reference


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Big bang/Critique of sample reference

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title: Big bang/Critique of sample reference


text:
Statistical Analysis/Unit 6 Content

Unit 6
This content is adapted from the Introduction to Statistics MA121/ECON104 Course at Saylor.org.

Linear Equations, Slope, and Y-intercept

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 12: Linear Regression and Correlation”
 * “Section 1: Linear Regression and Correlation”
 * “Section 2: Linear Equations”
 * “Section 3: Slope and Y-Intercept of a Linear Equation”

Instructions: Please read each of the linked sections above in their entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Scatter Plots

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 12: Linear Regression and Correlation"
 * "Section 4: Scatter Plots”

Instructions: Please read the entire webpage in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

The Regression Equation

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 12: Linear Regression and Correlation"
 * "Section 5: The Regression Equation”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Khan Academy’s Statistics
 * “Covariance and the Regression Line”

Instructions: Please view the video lecture in its entirety (approximately 15 minutes). This lecture will discuss covariance, variance, and the slope of the regression.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage displayed above.

The Correlation Coefficient

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 12: Linear Regression and Correlation"
 * "Section 6: The Correlation Coefficient”

Instructions: Please read the entire webpage in its entirety.

 * Lecture: Khan Academy’s Statistics
 * “Correlation and Causality”

Instructions: Please view the entire video lecture (10:25 minutes). This lecture will discuss correlation and causality.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage displayed above.

Prediction

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 12: Linear Regression and Correlation"
 * "Section 8: Prediction”

Instructions: Please read the entire webpage in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Outliers

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 12: Linear Regression and Correlation"
 * "Section 9: Outliers”

Instructions: Please read the entire webpage in its entirety.

Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 12: Linear Regression and Correlation”

Instructions: Please view the lecture to the right in its entirety.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 12: Linear Regression and Correlation”
 * “Practice: Linear Regression”

Instructions: Click on the hyperlinks titled “Practice: Linear Regression” and solve the problems. Next, click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21-25. The solutions are provided below the problem. Please solve all of the problems before checking the solutions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

ANOVA

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 13: F Distribution and ANOVA”
 * “Section 1: F Distribution and ANOVA”
 * “Section 2: ANOVA”

Instructions: Please read each of the linked sections in their entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

The F Distribution and the F Ratio

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 13: F Distribution and ANOVA”
 * “Section 3: The F Distribution and the F Ratio”
 * “Section 4: Facts About the F Distribution”

Instructions: Please read each of the linked sections above in their entirety.

 * Lecture: Khan Academy’s Statistics
 * “ANOVA 1 - Calculating SST (Total Sum of Squares)”
 * “ANOVA 2 - Calculating SSW and SSB (Total Sum of Squares within and between)”
 * “ANOVA 3 -Hypothesis Test with F-Statistic”

Instructions: Please view the three lectures, which discuss ANOVA, in their entirety (approximately 30 minutes total).

Terms of Use: Please respect the copyright and terms of use displayed on the webpages displayed above.

The F Distribution and the F Ratio

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 13: F Distribution and ANOVA"
 * "Section 5: Test of Two Variances”

Instructions: Please read the entire section.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 13: F Distribution and ANOVA”
 * “Practice: ANOVA”

Instructions: Click on the hyperlinks titled “Practice: ANOVA” and solve the problems. Next, click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7. The solutions are provided below the problem. Please solve all of the problems before checking the solutions.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 6 Content


text:
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$$a y' + b y^2 = \sin x + x^3$$

title: User:Egm6321.f10.team17.smith/R2


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text:
User:EGM6321.f09.team1.Zhichao Gong/Mtg4

Mtg 4: Tue, 1 Sep 09

page4-1

$$Redu{\color{blue}\underline} \ of \ {\color{blue}\underline} \ \phi \ :$$

$$M{\color{blue}\underline} \ able.$$

$${\color{red}\underline}$$

$$G({y}^{n}, \ {y}^{n-1}{\color{red}order \ n}, \ ... \, \ {y}^{''}, \ {y}^{'}, \ y{\color{blue}(dependent \ varible)}, \ x{\color{blue}(independent \ varible)}) \ = \ 0 \ {\color{red}(1)}$$

$$Missing \ y \ in \ G(.) \ :$$

$$G({y}^{n}, \ {y}^{n-1}, \ ... \, \ {y}^{''}, \ {y}^{'}, \ x) \ = \ 0 $$

$$Reduce \ order \ by \ $$

$$\Rightarrow$$

$$If \ we \ can \ solve \ (3) \ for \ p(x) \ then \ (2) \ \Rightarrow$$

page4-2

$${\color{blue}\underline{Application}}$$

$${y}^{''}+{y}^{'}=x \ {\color{red}(1)}$$

$$G({y}^{}, \ {y}^{'}, \ y \ x) \ = \ {y}^{}+{y}^{'}-x \ = \ 0 $$

$$let \ p(x)={y}^{'}(x) \ \Rightarrow \ {p}^{'}+p=x \ {\color{blue}linear \ first \ order \ ODE}$$

Use method ofintergate factor :

$$p(x)= \underset{ \overset{ \uparrow}}{A} {e}^{-x}+x-1 \ {\color{red}(2)}$$

$$y(x)=\int_{}^{x}p(z)dz=-\underset{ \overset{ \uparrow}}{A}{e}^{-x}+\frac{{x}^{2}}{2}-x+\underset{ \overset{ \uparrow}}{B}$$

HW: Derive Eq(2) using intergate factor.

Intergrate factor method

$$1st \ order \ ODEs \ (can \ be \ nonlinear)$$

$$\Rightarrow$$

$$M(x,y)+N(x,y){y}^{'}=0$$

M, N nonlinear wrt (x,y)

page4-3

$${\color{blue}\underline{Application}}$$

$$M(x,y)=2{x}^{2}+\sqrt{y}$$

$$N(x,y)={x}^{5}{y}^{3}$$

$${\color{blue} \underset{M(x,y)}{ \underbrace}}+{\color{blue} \underset{N(x,y)}{ \underbrace}}{\color{black}{y}^{'}=0} \ {\color{red}(1)}$$

HW: show that the 1st order ODE above is nonlinear


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User:Marshallsumter/Radiation astronomy/First X-ray source in Aquarius

The first X-ray source in Aquarius is unknown. The field of X-ray astronomy is the result of observations and theories about X-ray sources detected in the sky above. The first astronomical X-ray source discovered may have been the Sun. But, X-rays from the Sun do not penetrate far enough into Earth's atmosphere to be detected on the ground. The early use of sounding rockets and balloons to carry detectors high enough may have detected X-rays from the Sun as early as the 1940s.

This is a lesson in map reading, coordinate matching, and researching. It is also a research project in the history of X-ray astronomy looking for the first astronomical X-ray source discovered in the constellation of Aquarius.

Nearly all the background you need to participate and learn by doing you've probably already been introduced to at a secondary level.

Some of the material and information is at the college or university level, and as you progress in finding X-ray sources, you'll run into concepts and experimental tests that are actual research.

To succeed in finding an X-ray source in Aquarius is the first step. Next, you'll need to determine the time stamp of its discovery and compare it with any that have already been found. Over the history of X-ray astronomy a number of sources have been found, many as point sources in the night sky. These points are located on the celestial sphere using coordinate systems. Familiarity with these coordinate systems is not a prerequisite. Here the challenge is geometrical, astrophysical, and historical.

Traveling X-ray source
Many X-ray sources do not remain in a constellation for lengthy periods. Some of these are the Sun and sources apparently in orbit around the Sun. The Sun travels through the 13 constellations along the ecliptic, the 12 of the Zodiac and the constellation Ophiuchus.

Backgrounds
To introduce yourself to some aspects of the challenge may I suggest reading the highlighted links mentioned above, and if you're curious, those listed under the section "See also" below.

X-rays are a form of radiation that is currently part of the electromagnetic radiation intersecting the Earth. More information about radiation is in radiation astronomy.

Aquarius
The Wikipedia article about the constellation Aquarius contains a high school level description. The figure at right shows the sky map of Aquarius. Around the edges of the map are coordinates related to longitude and latitude, but with the Earth rotating on its axis every 24 hours the celestial coordinates must remain fixed. How has this been accomplished?

Also, in the Wikipedia article is a list of stars in Aquarius. What's the difference between a star and an astronomical X-ray source?

From the Wikipedia article on the Zodiac, Aquarius is a zodiacal constellation.

Searching catalogs
In the lecture/article X-ray astronomy in the Science section is a list of older catalogs of X-ray sources. Using the constellation description in the previous section, scan through coordinates for these X-ray sources to see if any may be within Aquarius.

If you find any that are, skip down to the section X-ray source in Aquarius and make an entry. Be sure to check the coordinate era, most B1950 coordinates have changed slightly to the new J2000 set. Try the catalog designation at either SIMBAD website.

X-1 designation
In the lecture/article first astronomical X-ray source, a number of X-ray sources carry a designation containing the notation "X-1". Searching Google scholar or other web browsers listed under External links below, using "X-1", with quotes, and "Aquarius", without quotes, may locate an early source. Adding the notation "X-rays" or "X-ray", either with quotes, may reduce the search options.

One astronomical object located by such a search is NGC 7293.

Testing a source
There are many web sites that may have an X-ray source listed for the constellation Aquarius. Some that you may wish to try are in the External links section near the bottom of this lesson.

Wikipedia sources
A. Constellation article

Under "Notable features" in the Wikipedia article on the constellation Aquarius is the list of stars in Aquarius. Click on this link. In the table of this Wikipedia article is β Aqr. To the right are coordinates:
 * 1) Right ascension (RA): 21h 31m 33.52s and
 * 2) Declination (Dec): -05° 34' 16.2".

Find these coordinates on the Aquarius map at the right. Is beta Aquarii really inside the boundaries of the constellation?

To evaluate the star as an X-ray source, skip ahead to section "X-ray source".

B. Wikipedia search

Another way to look for X-ray sources in the constellation is to perform a search on Wikipedia. Try "Aquarius "X-ray"" without the outside quotes. This yields about 25 returns which include EZ Aquarii, NGC 7252, 1RXS J210034.1-044442, AE Aquarii, Beta Aquarii, Supercluster, and Gliese 876.

To evaluate each of these as an X-ray source, skip ahead to section "X-ray source".

SIMBAD sources
Another way to find possible X-ray sources in Aquarius is to use search queries on SIMBAD.

Click on either SIMBAD link under "External links" below, then click on "Criteria query".

In the tan box, type in "region(21 31 33.52 -05 34 16.2, 10m)", without the quotes. This tells the SIMBAD computer you are interested in a circular region of the celestial sphere centered on the coordinates for beta Aquarii, with a radius of 10 arcminutes (m).

Notice on the page over at the right from the tan colored box: "Return". The default is "object count". Click on "submit query". In a few moments a result something like "Number of objects:	   	 4" should appear. Click "Back" to see the tan box again.

Add "otype='X'" to the entry so that it reads: "region(21 31 33.52 -05 34 16.2, 10m) & otype='X'", without the outside quotes. Click on "submit query" again. If the result is "0" number of objects, increase the number of arcminutes, or use "1d" which stands for "one degree".

When you find at least one object, change "Return" to "display" by clicking on the circle to its left, then "submit query".

SAO/NASA Astrophysics Data System
In the naming of sources per constellation, the genitive is in common use. For Aquarius, the genitive is Aquarii.

Click on the SAO/NASA Astrophysics Data System link below in the "External links". Try "Aquarii X-ray" without the quotes, or "alpha aquarii" with quotes, followed by "X-ray" without quotes.

Click on a link below # Bibcode Authors. If the Abstract describes the detection of X-rays from a source in the constellation Aquarius, go to the next section under "SAO/NASA Astrophysics Data System".

If it does not try another bibcode link.

X-rays
There are several ways to evaluate an X-ray source for the constellation Aquarius.

Wikipedia source
Click on the link to the Wikipedia article. After you've enjoyed reading about the source, use the 'find' command of your browser to see if this Wikipedia page mentions anything about "X-ray", or "X-rays". Does the article mention whether or not the source is an X-ray source?

What is the current time stamp for the Wikipedia article on the source? [Hint]: look for something like "This page was last modified on 25 December 2011 at 20:12." very near the bottom of the page. For now this is an adequate time stamp.

From reading the Wikipedia article on the source, if you believe the text demonstrates that the source is not an X-ray source in Aquarius edit the "Non-X-ray source in Aquarius" section near the bottom of the page with an entry similar to "# Alpha Aquarius 25 December 2011 at 20:12 Wikipedia article "Alpha Aquarius".", without the outer quotes, and finish the entry with four "~"s without the quotes after the period. The date included with your designation or username is a time stamp for the entry. The last portion of the entry is the source of your information.

On the other hand, if there are one or more sentences in the article that you believe demonstrates that the source is an X-ray source in Aquarius edit the section below "X-ray source in Aquarius" with a similar entry.

Go to the section entitled, "Challenging an entry".

SIMBAD source
To check any source (even one from Wikipedia) on SIMBAD, click of the "External link" to the "SIMBAD Astronomical Database".

At the lower right side of the SIMBAD Astronomical Database page is a "Basic search" box. There are several ways to try your target:
 * 1) source name: without the quotes or
 * 2) source coordinates: without the quotes, for example, "21 31 33.52 -05 34 16.2".

If you are looking at a SIMBAD generated table which lists possible targets, click on one.

On its SIMBAD page read down the left side until you see "Other object types:". To the right of this is a list of other object types that the source is. Look for an X. Is there one in this horizontal list?

If an X is present, skip down to the section, "SIMBAD time stamp". Even if an X is not present, noting that SIMBAD does not consider the source to be an X-ray source is important, so skip down to the "SIMBAD time stamp" section.

If you have already found an X-ray source (or a table of them) using SIMBAD, click on the blue link identifier for the first.

SIMBAD time stamp
Peruse the SIMBAD page for a time stamp or date of last revision. [Hint: it may look something like "2012.01.09CET20:10:02" and be in the upper right.]

If the entry at SIMBAD convinces you that the source is not an X-ray source, edit the "Non-X-ray source in Aquarius" section near the bottom of this page and type in an entry similar to "# Source Name 2012.01.09CET20:10:02 SIMBAD article "SIMBAD source name".", without the first set of quotes, followed by four ~s.

If your SIMBAD analysis convinces you that you have found an X-ray source in Aquarius (did you check the coordinates vs. the map of Aquarius?), make an entry something like the ones in the section "X-ray source in Aquarius".

SAO/NASA Astrophysics Data System source
If the abstract states that an X-ray source in Aquarius is detected or studied, consider entering it in the section "X-ray source in Aquarius" below.

Abstract time stamp
On the abstract page is a Publication Date:. This may serve as a time stamp for establishing that the source is detected as an X-ray source on or before the date of publication. The time stamp followed by four ~s for your verification as determiner in the section "X-ray source in Aquarius" completes your entry.

Challenging an entry
Any entry in either the section "X-ray source in Aquarius" or "Non-X-ray source in Aquarius" can be challenged. The time stamp can be challenged to see if there is an earlier one. The source can be challenged by an earlier source.

Wikipedia challenge
Is Wikipedia a 'primary source', or does the Wikipedia article cite a source?

Even though Wikipedia has an article on the source, is it a good place to stop in testing whether the source has been detected as an astronomical X-ray source?

If the Wikipedia article cites a primary source, skip down to the section on "Primary sources".

SIMBAD challenge
Is SIMBAD a 'primary source'?

SIMBAD is an astronomical database provided by the Centre de Données astronomiques de Strasbourg. It is an authoritative source, but they do occasionally make a mistake.

If you find an X-ray source within the constellation on SIMBAD, the next step is to find the earliest time stamp of discovery.

SAO/NASA Astrophysics Data System challenge
Is the SAO/NASA Astrophysics Data System abstract entry a primary source?

The SAO/NASA Astrophysics Data System is an astronomical database provided by the High Energy Astrophysics Division at the Harvard-Smithsonian Center for Astrophysics of Harvard University. The abstract has been copied from the actual article in a scientific journal or other publication. Mistakes can be made and the article may record within its text exact dates when the observation or detection of X-rays actually occurred. Such a record may provide an earlier time stamp.

Wikipedia test source
For a Wikipedia article that cites a primary source, scroll down to the reference and open the reference. Read through the article looking for where the source mentioned in the Wikipedia article occurs. Some primary source authors may use source designations that are not mentioned in the Wikipedia article. To look for other designations, click on the link to SIMBAD in the "External links" on this page, enter the source name from the Wikipedia article, and see if other names are mentioned in the article.

When none of the names are mentioned, click on the link for "Google Advanced Search" in the list of "External links", enter the source name or designation(s) such as "Gliese 866", with "X-ray" to see if the source has a reference indicating it is an X-ray source. And, look for the earliest one. Compose an entry using the primary source.

SIMBAD test source
Further down the SIMBAD page is a list of "Identifiers". Click on the blue bold portion.

On the page that appears should be a primary source listed after Ref:. Click on the blue link with the oldest year. This yields an earlier time stamp and entry citation like the current one in the section "X-ray source in Aquarius". If you find another source or an earlier time stamp, compose a similar entry and edit the section. Additional information to add into the reference can be found by clicking on "ADS services" from the SIMBAD page.

SAO/NASA Astrophysics Data System source test
Click on either the "Electronic Refereed Journal Article (HTML)" or "Full Refereed Journal Article (PDF/Postscript)", if available.

Depending on the article display, if the abstract is repeated and the article is listed as FREE, click on either the PDF or HTML version.

While scanning or reading the article look for "Observations" (or use the Find function of your browser) and the possible inclusion of dates for these. If more than one X-ray source in Aquarius are detected, which one(s) would you list in the section "X-ray source in Aquarius" below?

An example of an article reference is provided in that section.

Changing an entry
From your analysis of the source so far, is it an X-ray source?

If you have found an earlier time stamp for the source than the one listed in the section below "Non-X-ray source in Aquarius" and the answer to the above question is "no", you can edit the section with your result. Or, you can leave the entries as is and try another star.

If you have found an earlier time stamp for the source than the one listed in the section below "X-ray source in Aquarius", edit the section with your result. Or, if you found another X-ray source with a comparable or earlier time stamp, edit the section with your result.

X-ray source in Aquarius

 * 1) EZ Aquarii March 1984.  Marshallsumter 21:06, 12 January 2012 (UTC).
 * 2) 1RXS J213145.0-055439 1999 SIMBAD article "1RXS J143729.3-791432" Marshallsumter 22:30, 12 January 2012 (UTC).
 * 3) EZ Aquarii 30 November 2011 at 04:55 Wikipedia article "EZ Aquarii" Marshallsumter 19:41, 12 January 2012 (UTC).
 * 4) beta Aquarii 28 December 2011 at 20:06 Wikipedia article "Beta Aquarii" Marshallsumter 22:43, 12 January 2012 (UTC).
 * 5) 1RXS J213145.0-055439 2012.01.12CET23:10:30 SIMBAD article "1RXS J213145.0-055439" Marshallsumter 22:30, 12 January 2012 (UTC).
 * 6) alpha Aquarii 17 January 2005 for 18.20 ks. Marshallsumter (talk) 06:30, 13 June 2012 (UTC).
 * 7) NGC 7252, "X-ray emissions were observed in NGC 7252. This suggests the existence of nuclear activity or an intermediate-mass black hole in the galaxy." Wikipedia article on NGC 7252, November 15, 2010, at 02:37. --Marshallsumter (discuss • contribs) 06:05, 10 May 2013 (UTC). Observation with the X-ray satellite ASCA of "a nearby merger remnant NGC 7252 ... detected X-ray emission with the X-ray flux of (1.8 ± 0.3) x 10-13 ergs s-1 cm-2 in the 0.5-10 keV band. ... hard X-ray emission ... may indicate the existence of nuclear activity or an intermediate-mass black hole in NGC 7252." --Marshallsumter (discuss • contribs) 14:43, 10 May 2013 (UTC). Using the preceding article as a source, an earlier X-ray observation has been found. "The ROSAT PSPC observations were obtained in 1992 December and 1993 January and amount to 17 381 s of data centered on NGC 7252." --Marshallsumter (discuss • contribs) 15:30, 10 May 2013 (UTC).
 * 8) 4U 2305-07 from "The fourth Uhuru catalog of X-ray sources" is within the figure for the constellation Aquarius. According to the lecture/article X-ray astronomy, "Sources detected during the final observation period from August 27, 1973, to January 12, 1974, are prefixed with "4U"." --Marshallsumter (discuss • contribs) 16:59, 10 May 2013 (UTC). SIMBAD at 2013.05.10CEST19:00:06 claims 4U 2305-07 is Markarian (Mrk) 926, a Seyfert type 1 galaxy. --Marshallsumter (discuss • contribs) 17:14, 10 May 2013 (UTC). This is the only X-ray source in the 4U catalog within Aquarius. --Marshallsumter (discuss • contribs) 21:57, 10 May 2013 (UTC).
 * 9) The "X-1" Aquarius "X-rays" search on Google scholar came up with "Some examples are NGC 7293 (Helix) in the Aquarius constellation, NGC 7027, and NGC 7009. ... coming from this collisional process ionizes and heats up the gas and may eventually be converted into radiation, including X-rays." SIMBAD at 2013.05.10CEST20:01:49 lists NGC 7293 as an X-ray source detected by 2E (Einstein Observatory), 2RXP (ROSAT), and 2XMM (XMM Newton). --Marshallsumter (discuss • contribs) 21:13, 10 May 2013 (UTC).
 * 10) Also, from the "X-1" Aquarius "X-rays" search is "Her X-1, as well as the cataclysmic variables AM-Herculis and AE-Aquarius (Weekes, 1988". SIMBAD at 2013.05.11CEST00:53:50 produces a record when "Aqr X-1", without the quotes, is input for a basic search: "X Aqr X-1 -- Cataclysmic Binary Candidate", but without coordinates or other designations. SIMBAD at 2013.05.11CEST01:00:11 yields using AE Aquarii: "V* AE Aqr -- CV DQ Her type (intermediate polar)" otype='X': (1AXG,1E,2E,1ES,1RXS,2XMM,[BM83],[FS2003]). --Marshallsumter (discuss • contribs) 23:03, 10 May 2013 (UTC).

Non-X-ray source in Aquarius

 * 1) beta Aquarii 2012.01.12CET23:31:47 SIMBAD article "* bet Aqr".
 * 2) alpha Aquarii 27 January 2012 at 04:04 Wikipedia article Alpha Aquarii. Marshallsumter (talk) 00:37, 4 March 2012 (UTC)
 * 3) alpha Aquarii 2012.03.04CET01:36:36 SIMBAD article "NAME SADALMELIK". Marshallsumter (talk) 00:40, 4 March 2012 (UTC)
 * 4) NGC 7252 2013.05.10CEST07:52:57 SIMBAD article "NGC 7252". --Marshallsumter (discuss • contribs) 06:05, 10 May 2013 (UTC)

Oldest record
4U 2305-07 from "The fourth Uhuru catalog of X-ray sources" is within the figure for the constellation Aquarius. The article dates from December 1978. According to the lecture/article X-ray astronomy: "Sources detected during the final observation period from August 27, 1973, to January 12, 1974, are prefixed with "4U"."

SIMBAD annotations
For NGC 7252, the SIMBAD record indicates that it is not an X-ray source yet above at least two refereed journal articles indicate that it is. Use the second SIMBAD External links to directly display the SIMBAD database in France.

Enter "NGC 7252", or the source you have found, without the quotes into the search box. Scroll down to the Annotations :. Look for the link "add an annotation to this object". With access to the literature citations available, click on this link. You may need to register as a user. It's free. Post your annotation containing the literature references.

Aquarius X-1
SIMBAD at 2013.05.11CEST00:53:50 produces a record when "Aqr X-1", without the quotes, is input for a basic search: "X Aqr X-1 -- Cataclysmic Binary Candidate", but without coordinates or other designations.

A Google scholar search using "Aqr X-1", with quotes produces one source. In its Table 3 entitled: "Table of observed number of upward showering muons and expected background from atmospheric neutrinos in a cone of half-angle 3◦ from selected point sources along with 90% c.l. muon and neutrino flux limits for a E−2 neutrino spectrum." , is an entry "Aqr X-1". No coordinates, any other designation, or literature reference is given.

In examining the article entitled, "33 second X-ray pulsations in AE Aquarii", there is the designation X-1 referring to their first X-ray observation run using the Einstein Observatory. AE Aqr is at J2000 RA 20h 40m 09.1629s Dec -00° 52' 15.071". This X-1 X-ray run has an X-1A and an X-1B, followed by an X-2 observation run. This may have led to AE Aquarii being thought of as Aquarius X-1. AE Aquarii is a cataclysmic variable with nova-like activity and an orbital period of 9.88 hours. "A K dwarf fills its critical Roche surface and transfers matter to a white dwarf via an accretion disk." AE Aquarii was observed "for a total of 1.3 hours on 1979 April 27 and 28." No earlier X-ray observations were mentioned.

Hypotheses

 * 1) The first X-ray source in Aquarius was detected by Ariel 5.


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User:Khateeb88

Firas Alkhateeb

I am a high school History teacher in Chicago, IL. I teach world, U.S., and Islamic history.

I am a native speaker of English and Arabic, and am currently learning Turkish. I will try to contribute to the Turkish language learning pages when I can.

Knowledge is power.

title: User:Khateeb88


text:
User talk:Khateeb88

 Hello Khateeb88, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --HappyCamper 04:45, 12 January 2012 (UTC)

title: User talk:Khateeb88


title: File:Iea.s12.sec1.djvu


text:
Talk:Hydraulic tables and chairs design

This article is, once again, a nonsense. There are several problems:

 * Cost. Hydraulics costs money. A dentist's chair has one short hydraulic ram for lift (Tilt is usually a leadscrew instead, although some US chairs use hydraulics here too, because of patient weight). Such chairs cost thousands and also requires occasional servicing. How does a table with multiple long rams end up as cheap furniture for people in tiny apartments?

 * Space underfloor. Unlike the many mechanical linkages used for tables, hydraulic rams don't fold up. They require even more space under the floor than their height above it. This might be available in bungalows, but not in multi-storey apartments where height is even more costly than floor area. Multi-section telescopic rams reduce this height needed, but they still need substantial height and they add cost and complexity.

 * A third problem is the simple difficulty of doing this with multiple rams. There are many long thin rams here, and any tilt will lead to jamming. Making this work without stiction would be near-impossible,

There have been many folding table in the past and no doubt more in the future. However they'll use cheap, compact linkages, not multiple hydraulic rams. Andy Dingley 13:20, 12 January 2012 (UTC)

title: Talk:Hydraulic tables and chairs design


text:
User:Areil314/sandbox/sample3

(Review Paper) Cited in

title: User:Areil314/sandbox/sample3


text:
Big bang/A Critique of ASTROPHYSICS: Precision Cosmology? Not Just Yet

(Review Paper) Cited in

relies on theoretical forces
The Big bang theory requires dark energy to exist. Dark energy is a theoretical force, we can't actually measure it.


title: File:C.Repeat.Loop.1.A.20120112.pdf


text:
TESOL/Agreeing and disagreeing

Review
{Put these words in the correct order. + Word 1 ---+- Word 2 -+--- Word 3 + Word 4 --+-- Word 5
 * type=""}
 * well | but | necessarily | perhaps | not

{Put these words in the correct order. -+ Word 1 ---+-- Word 2 +- Word 3 --+--- Word 4 -+ Word 5 +- Word 6
 * type=""}
 * them | I | graduated | couldn't | have | without

{Put these words in the correct order. --+- Word 1 +--- Word 2 ---+ Word 3 +--- Word 4 -+-- Word 5 -+-- Word 6 --+- Word 7 ---+ Word 8
 * type=""}
 * me, | video | are | healthy | you | games | if | ask

title: TESOL/Agreeing and disagreeing


text:
User:Dets65

Hi!

title: User:Dets65


text:
User:EGM6321.f09.team1.Zhichao Gong/Mtg6

Mtg 6: Tue, 3 Sep 09

page6-1

Reverse engineering to invert billions and billions of exact nonlinear 1st order ODEs:

$$Just \ invert \ any \ \phi(x,y) \ {\color{blue}(nonlinear \ wrt \ (x,y))}$$

$$\phi(x,y)=6{x}^{4}+2{y}^{\frac{3}{2}}$$

$${\color{blue}p.5-3} \ {\color{red}Eq(1):} \ F=\frac{d}{dx} \phi(x,y(x))$$

$$ \begin{cases} & M={\phi}_{x} \\ & N={\phi}_{y} \end{cases}$$ $$\Rightarrow$$ $$M+N{y}^{'}=0$$ HW: Complete details and invent 3 more

page6-2

What if exactness condition 2 (p.6-1) not satisfied ? Euler intergate factor method: Assume p.5-2 Eq(2) not exact, then  mult it w/ h(x,y) yetr unkonwn(to
be determined)

$$h(x,y)[M(x,y)dx+N(x,y)dy]=0$$

$$\Rightarrow \ {\color{blue} \underset{ \overset{-}{M}}{ \underbrace}}dx+{\color{blue} \underset{ \overset{-}{N}}{ \underbrace}}dy=0 \ {\color{red}(1)}$$

Q: Find h (Euler intergrate factor) st

Eq(1) is exact.

Note: (1) already satisfiesexcatness condition 1 on p.6-1:

$${\color{blue}p.5-4} {\color{red}Eq(1):} \ { \overset{-}{M}}_{y}={ \overset{-}{N}}_{x}$$

page6-3

$${ \overset{-}{M}}_{y}={(hM)}_{y}={h}_{y}M+h{M}_{y}$$

$${ \overset{-}{N}}_{x}={(hN)}_{x}={h}_{x}N+h{N}_{x}$$

$$\Rightarrow$$

Solve Eq.(1) for h(x,y). Not easy in general.

2 Particular cases:

$${\color{blue} \underline{Case \ 1:}} \ If \ ({h}_{y}M)=0, \ (1) \ because$$

$${h}_{x}N+h({N}_{x}-{M}_{y})=0$$

$$\Rightarrow \ \frac{{h}_{x}}{h}=-{\color{red} \underset{known}{ \underbrace}}$$

$$ \underset{logh}{ \underbrace{\int \frac{{h}_{x}}{h}dx}}=-\int \frac{1}{N}({N}_{x}-{M}_{y})dx$$

page6-4

$${\underline{Note:}} \ {h}_{y}{\color{blue} \underset{ \neq0}{\underbrace}}=0 \ \Rightarrow \ {h}_{y}=0 $$

$$\Rightarrow$$

$$\Rightarrow$$

If (2) satisfied, then:

$${\color{blue}carl \ Sagan.}$$


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Turkish/Plural

Learning Goals
This lesson will teach you how to make nouns in Turkish plural.

Making Them
In English, you add "-s" usually at the end of a word to make it plural. In Turkish, we add either a -ler or -lar at the end of a noun to make it plural.

Examples:

gün (day) → günler (days)

akşam (evening) → akşamlar

gece (night) → geceler

televizyon (T.V.) → televizyonlar

-ler or -lar?
Since there are two ways to make things plural, we have to decide which one to chose for any given word. We do this by choosing the one that rhymes best with the last vowel in the word.

-ler works best with i, e, ö, and ü -lar works best with ı, a, o, and u

title: Turkish/Plural


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Finite elements/Differential form of physics

Here, we will give a short introduction to the partial differential equations that govern the basic conservation laws. By constructing these laws in differential form, we can learn more about how they will affect material in a control volume.


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User:Silentcivilian~enwikiversity

I will be updating this page soon.

title: User:Silentcivilian~enwikiversity