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# S301 Week 9 Lecture & Textbook Notes STAT-S301

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This 5 page Class Notes was uploaded by Lauren Detweiler on Wednesday March 25, 2015. The Class Notes belongs to STAT-S301 at Indiana University taught by Hannah Bolte in Spring2015. Since its upload, it has received 114 views. For similar materials see Business Statistics in Statistics at Indiana University.

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Date Created: 03/25/15

S301 Textbook Week 9 Ch 153155 amp161162pgs 366375 385398 Required concepts not covered in lecture 1 Common Confusions in Interpreting Con dence Intervals CIs F mrmm ml 392 A a c o c r 5 H n J K 1 so 2 L An Excelscrecn 35 m 239 20 15 12 m mov Count I 19 Balance 5 A picture from the textbook pg 367 The 95 CI is shown by the two red lines a Ninety ve percent of all customers keep a balance of 1520 to 2460 The con dence interval doesn39t contain many data much less 95 of the balances The con dence interval gives a range for the population mean u not the balance of an individual b The mean balance of 95 of samples of this size will fall between 1520 and 2460 The con dence interval describes u an unknown constant not the means of other samples c The mean balance 11 is between 1520 and 2460 Closer but still incorrect The average balance in the population does not have to fall between 1520 and 2460 This is a 95 con dence interval It might not contain u 11 Margin of Error a The 95 CI for u replaces the percentile from the t distribution with 2 S S x 2 W to x 2 W b Margin of Error ME the extent of this interval to either side of X or similarly around 23 S ME 2 W e Three factors determine the Margin of Error i Level of con dence ii Variation of the data iii Number of observations III Hypothesis Testing a dlevel probability of a Type I error b C the critical value for the sample mean X between the retain and reject regions c pvalues smallest alevel at which H0 can be rejected d Power the probability that a test can reject H0 IV The relationship between errors alevels C pvalues power a See above V z test for proportions and 190 a z statistic Number of standard errors that separate the test statistic from the region speci ed by H0 deviation of test statistic from H0 15 190 standard error of test statistic p01 pOn b z test test of H0 based on a count of standard errors separating HO from the test statistic Required concepts covered in lecture 1 Choosing the Sample Size a Suppose I want a 95 con dence interval but I want the CI to have some particular margin of error b Rearrange the equation for margin of error to solve for n the sample size 402 n Margin of Error2 c We have to choose n before doing anything i If you don t know a obtain an estimate using a pilot sample 11 Determining Sample Size for Proportions a For a study about p no need for a pilot sample b Use p 05 which results in largest possible values for a jp1 p V05 0 025 05 c Sample sizes for various margins of error 95 coverage for p 100 10 400 5 625 4 11 12 3 2500 2 10000 1 III From Con dence Intervals to Hypothesis Testing a Statistics uses observed data to make informed suppositions about a population i Confidence intervals provide a range of plausible values for a population parameter ii Hypothesis tests consider the plausibility of a speci c claim claims are called hypotheses 1 H0 Hnaught Hnull null hypothesis 2 Ha Sometimes written as H1 alternative hypothesis IV Attributes of the Null Hypothesis H0 a Gets the bene t of the doubt b Associated W no change in course c The default belief or status quo d Origination httpenWikipediaorgWikiLadytastingtea V Examples of the Null Hypothesis HO a In practice the null hypothesis is exempli ed When I Fisher presumed the lady couldn t tell the difference unless she adequately convinced him otherwise I Without evidence of cheating I presume you have your own choker I We don t x the widget machine unless we have adequate reason to believe it is malfunctioning I Those accused of a crime are innocent unless a jury of peers is convinced of their guilt beyond a reasonable doubt VI Attributes of the Alternative Hypothesis Ha a Requires us to take some action b Change our action c Change our belief d Carries the burden of proof e Is the only other option VII Examples of Null and Alternative Hypotheses a In hypothesis testing the null and alternative hypotheses must be mutually exclusive and exhaustive b Either the null is true or the alternative is true I NULL The tea drinker cannot distinguish when the milk was added I ALTERNATIVE The tea drinker can distinguish when the milk was added I NULL You are not answering clicker questions for someone else I ALTERNATIVE You are answering clicker questions for someone else I NULL The widget machine is ne I ALTERNATIVE The widget machine is not ne I NULL You are innocent of the crime I ALTERNATIVE You are guilty of the crime VIII Test Statistics a Statistical tests rely on the sampling distribution of the test statistic that estimates the parameter specified in the null and alternative hypotheses P P0 x L Z VP01 Pon or tn l S 2 b The test statistic tells how many standard errors your sample mean is from the expectation under the null 43 S S 9 9 45 44 r RAW 3 tmre quot r ip 1 1 4 go 39 2 1 5 z uort 2 test shrh39s s39c c Key Question What is the chance of getting a test statistic this far from the center of the null distribution if the null is true VI Hypothesis Testing dLevels and pValues a To test your hypothesis i Specify your threshold for reasonable doubt This is your oclevel ii Identify the critical value for your chosen oclevel between retaining and rejecting the null 0H 9 44491 Test Lefttailed 2 vs lt Right tailed S vs gt 2tailed 2 vs 7i Sign i Proportion za Za 2 or a known 0 unknown lam 1 tafZJI l iii Collect your data Identify your sample statistics 3 and S or 23 iv Calculate your test statistic tn1 or z V Consider probability of observing test statistic under null hypothesis 1 If test statistic is more extreme than critical value reject H0 in favor of Ha Otherwise retain H0 2 Equivalent to nding the probability of observing test statistic under the null is lt 0L we call this the pvalue 3 Equivalent to having observed a value for X that exceeds the critical value between the retain and reject region we call this value C VII Type I and Type II Errors a But what if I m wrong i When making a decision two things can go wrong 1 You reject the null when it s really true A Type I error 2 You retain the null when it s really false A Type 11 error a Type I errors are worse than Type II errors Retain the null Reject the null Null is really true Type I Error J Null is really false Type H Error indicates a correct decision VIII Type I Errors a You changed your belief took action stated something was meaningful when it wasn t b Worst type of error we can make c We limit our probability of making a Type I error to CL d gt quot The oclevel chosen is our willingness to commit a Type I error IX Type 11 Error a Rejecting the null when you shouldn t b You erred on the side of caution and hence failed to act on an opportunity or take action accept a new treatment s effect or accept something isn t working the way it should c A good test has enough power to reject the null when it is false X dLevels C pValues Power NULL mSY th wm gal 11 1 f 1 r 39 I l I I I I l x W 1w d I mm L I II I X I I w n RETAIN NULL C R szlc v C What happens to our probability of Type I error if we increase 0L What happens to C if we increase CL C What happens to power if true unknown distribution is very close to null distribution d What happens to probability of Type 11 error if our test has little power to reject null e What happens to our power if we increase n PT

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