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# 63 Class Note for STAT 30100 with Professor Gundlach at Purdue

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Date Created: 02/06/15

Chapter 7 Section 71 Inference for the Mean of a Population Section 72 Comparing Two Means Learning goals for this chapter Understand what inference is and why it is needed Know that all inference techniques give us information about the population parameter Explain what a confidence interval is and when it is needed Calculate a con dence interval for the population mean when the population standard deviation is unknown Know the assumptions that must be met for doing inference for the population mean when the population standard deviation is unknown robustness for 1 sample mean matched pairs and 2sample comparison of means Know how to write hypotheses calculate a test statistic and Pvalue and write conclusions in terms of the story Draw Normal curve pictures to match the hypothesis test Understand the logic of hypothesis testing and when a hypothesis test is needed Use the con dence interval to perform a twosided hypothesis test Explain sampling variability and the difference between the population mean and the sample mean Explain the difference between the population standard deviation and the sample standard deviation Know which technique is most appropriate for a story con dence interval hypothesis test or simple summary statistics Know which inference technique is most appropriate for a story lsample mean using Z lsample mean using t matched pairs or 2sample comparison of means Interpret Normal quantile plots and histograms to determine whether the t procedures are appropriate Know how to do all calculations listed above by hand with the t table and using SPSS In Chapter 6 we knew the population standard deviation 039 039 Con dence interval for the population mean u x i Z X n f7 0 UJ Hypothesis test statistic for the population mean u 20 Used the distribution 2 N041 J In Chapter 7 we don t know the population standard deviation 039 Use the sample standard deviation s 4 Con dence interval for the population mean y f it gtlt sxZ Hypothesis test statistic for the population mean u to t distribution uses n l degrees of freedom Sometimes you ll see the symbol for standard error 6 n Using the t distribution Suppose that an SRS of size n is drawn from aNu 0 population There is a different t distribution for each sample size so tk stands for the t distribution with k degrees of freedom Degrees offreedom k n 7 1 sample size 7 1 As k increases the t distribution looks more like the normal distribution because as 71 increases 3 gt 039 tk distributions are symmetric about 0 and are bell shaped they are just a bit wider than the normal distribution Table shows upper tails only so o ift is negativePt lt t Pt gt t U o if you have a 2sided test multiply the Pt gt t by 2 to get the area in both tails o The Normal table showed lower tails only so the t table is backwards Finding t on the table Stalt at the bottom line to get the right column for your confidence level and then work up to the correct row for your degrees of freedom What happens if your degrees of freedom isn t on the table for example df 79 Always round DOWN to the next lowest degrees of freedom to be conservative Example afcan dence mtsmzfmtnmntnss An agncultural expert perfums a study tu measureyxeld ufaturnatu eld Studyrng 1n pluts ufland she dndsthe rnean yreld is 34 bushels wrth asample standard demauun uf 12 75 and a 95 cun dence rnterval fur the unknuwn nunulauun rnean yreld ufturnatues Example nftestfmtnmntnss Cunduet ahyputhesrs test wrth 11 u as tu detamxnexfthe pupulanun rnean yreld uftumatuesxsless than 42 bushels State yuurcunclusmnm terrns ufthe stury Alsu draw apxcture ufthet eurve wrththe nurnherand symhnl furthe pupulanun rnean yuu use m yuur nun hyputhesxs ya the sample rnean m the standard s enur a r M n and the test statrsue 1n Alsu shade the annrunnate pan ufthe J eurve whreh shuws the Prvallle Example Exercise 737 How accurate are radon detectors of a type sold to homeowners To answer this question university researchers placed 12 detectors in a chamber that exposed them to 105 picocuries per liter of radon The detector readings were as follows 919 978 1114 1223 1054 950 1038 996 1193 1048 1017 966 a Is there convincing evidence that the mean reading of all detectors of this type differs from the true value of 105 Use or 010 for the test Carry out a test in detail and write a brief conclusion SPSS tells us the mean and standard deviation of this data are 10413 and 940 respectively b Find a 90 confidence interval for the population mean Now redo the above example using SPSS completely To do just a con dence interval enter data then Analyze 9Descriptive Statistics 9 Explore Click on Statistics anal change the CI to 90 Then hit OK If you need to do a hypothesis test anal a CI go to Analysis 9Compare Means 9 One sample T test Change the test value to 105 since that is our Hg change options to 90 and hit OK This will give you the output below OneSample Test Test Value 105 90 Confidence Interval of the Difference Mean t df Sig 2tailed Difference Lower Upper radon detector readings 319 11 755 8667 5739 4005 Usmg ths SFSS EIutpuL omsmm Ya Yes Valuezm am Can denue mevva a he Dmevenue Mean m Swg muse Dmerence Lava Uppev 1mm datedmveawrgs 7319 M 755 75557 5 739 3 ms what wuma yuurtrcurve mm shaded Prvalue luuk hke xfyuu had hypumeses uf Hn 41n5 HA Hgms Hn wlns HA wms Hn nilEIS HA ms Yuu must chuuse yuurhyputheses BEFORE yuu examme the data When m auum an a twursxdedtest How do you know when it is appropriate to use the t procedures Very important Always look at your data rst Histograms and Normal quantile plots pgs 8083 in your book will help you see the general shape of your data I t procedures are quite robust against nonnormality of the population except in the case of outliers or strong skewness 0 larger samples n improve the accuracy of the t distribution Some guidelines for inference on a single mean I n lt 15 Use t procedures if data close to normal If data nonnormal or if outliers are present do not use t I 1557 540 Use t procedures except in the presence of outliers or strong skewness I n 2 40 Use t procedures even if data skewed Normal quantile plots In SPSS go to Graphs QQ Move your variable into variable column and hit OK Normal QQ Plot of Radon Detector Reading 120 o I 100 Expected Normal Value 90 l I I 90 100 MO 120 Observed Value Look to see how closely the data points dots follow the diagonal line The line will always be a 45degree line Only the data points will change The closer they follow the line the more normally distributed the data is What happens if the t procedure is not appropriate What if you have outliers or skewness with a smaller sample size n lt 40 Outliers Investigate the cause of the outliers 0 Skewness Was the data recorded correctly Is there any reason why that data might be invalid an equipment malfunction a person lying in their response etc If there is a good reason why that point could be disregarded try taking it out and compare the new con dence interval or hypothesis test results to the old ones If you don t have a valid reason for disregarding the outlier you have to leave the outlier in and not use the t procedures If the skewness is not too extreme the t procedures are still appropriate if the sample size is bigger than 15 Ifthe skewness is extreme or ifthe sample size is less than 15 you can use nonparametric procedures One type of nonparametric test is similar to the t procedures except it uses the median instead of the mean Another possibility would be to transform the data possibly using logarithms A statistician should be consulted if you have data which doesn t t the t procedures requirements We won t cover nonparametric procedures or transformations for nonnormal data in this course but your book has supplementary chapters 14 and 15 on these topics online ifyou need them later in your own research They are also discussed on pages 4654 70 of yourbook What do you do when you have 2 lists of data instead of 1 First decide whether you have 1 sample with 2 measurements on each unit OR 2 independent samples with one measurement each l Matched Pairs covered in 71 0 One group of individuals with 2 different measurements on each individual 0 Same individuals different measurements 0 Examples pre and posttests before and after measurements 0 Con dence intervals and hypothesis tests are based on the difference obtained between the 2 measurements 1 Find the difference post test pre test or before after etc in the individual measurements 2 Find the sample mean a and sample standard deviation s of these differences 3 Use the t distribution because the standard deviation is estimated from the data 0 Confidence interval 57 if ulh I Hypothesis testing Hg may 0 7 ttest statistic39 t I 0 sxZ Example of Matched Pairs In an effort to determine whether sensitivity training for nurses would improve the quality of nursing provided at an area hospital the following study was conducted Eight different nurses were selected and their nursing skills were given a score from 110 After this initial screening a training program was administered and then the same nurses were rated again Below is a table of their pre and posttraining scores Conduct a test to determine whether the training could on average improve the quality of nursing provided in the population individuals Pretraining Posttraining score score 1 256 454 2 322 533 3 345 432 4 555 745 5 563 700 6 789 980 7 766 533 8 620 680 a What are your hypotheses b What is the test statistic c What is the Pvalue d What is your conclusion in terms of the story e What is the 95 confidence interval of the population mean difference in nursing scores Enter the pre and post training scores to SPSS Then Analyze Compare Means Paireal Samples T test Then input both variable names and hit the arrow key If you need to change the con dence interval go to Options SPSS will always do the left column of data 7 the right column of data for the order of the ali erence If this bothers you just be careful how you enter the data into the program Paired Samples Statistics Std Error Mean N Std Deviation Mean Pair Posttrainin 9 score 63212 8 182086 643 77 1 Pretraining score 52700 8 201808 71350 Data entered as written above with pretraining in left column and posttraining in right column Paired Samples Test Sig 2 Paired Differences t df tailed Std Std Error 95 Confidence Interval Mean Deviation Mean of the Difference Lower Upper Pa pretra39rl39hg 39 405125 147417 52120 228369 18119 2017 7 084 posttralnlng Data entered backwards from how it is written above with posttraining in left column and pretraining in right column Paired Samples Test Paired Differences 95 Con dence Interval ofthe Std Error Difference Mean Std Deviation Mean Lower Upper t df Sig 2tailed a39r Pgst39t39a39quot39quotg we 105125 147417 52120 18119 228369 2017 084 retralnlng SCOI39E What s different What s the same Which one matches the way that you defined MW 10 2 2 Sample Comparison of Means covered in 72 I A group of individuals is divided into 2 different experimental groups 0 No one unit can be in both groups Each individual receives only one treatment andor is measured only once 0 Responses from each sample are independent of each other 0 Examples treatment vs control groups male vs female 2 groups of different women 0 Goal To do a hypothesis test based on Hg yA 3 same as Hg uA y 0 Ha yA gt y or Ha yA lt LIB or Ha uA yB pick one 2 Sample t Test Statistic is used for hypothesis testing when the standard deviations are ESTIMATED from the data these are approximately I distributions but not exact 0 Confidence Interval for M LIB 2 2 EA ithdfmin nA 1725 l nil n5 Equal sample sizes are recommended but not required Use the same guidelines for determining whether the t procedures are appropriate that you used for lsample mean and matched pairs but use 71 m m for the sample size ll Example of 2Sample Comparison of Means A group of 15 college seniors are selected to participate in a manual dexterity skill test against a group of 20 industrial workers Skills are assessed by scores obtained on a test taken by both groups Conduct a hypothesis test to determine whether the industrial workers had signi cantly better average manual dexterity skills than the students Descriptive statistics are listed below Also construct a 95 con dence interval for this problem group n x s students 15 3512 431 workers 20 3732 383 Example of 2Sample Comparison of Means Exercise 7 84 The SSHA is a psychological test designed to measure the motivation study habits and attitudes towards learning of college students These factors along with ability are important in explaining success in school A selective private college gives the SSHA to an SRS of both male and female rstyear students The data for the women are as follows 154 109 137 115 152 140 154 178 101 103 126 126 137 165 165 129 200 148 Here are the scores for the men 108 140 114 91 180 115 126 92 169 146 109 132 75 88 113 151 70 115 187 104 a Test whether the population mean SSHA score for men is different than the population mean score for women State your hypotheses carry out the test using SPSS obtain aPvalue and give your conclusions When you enter your data into SPSS have 2 variables gender type string anal score numeric In the gender column state whether a score is from a man or a woman anal in the score column state all 38 scores Analyze 9Compare Means 91ndepenalent Samples T Test Move score into Test Variables box Move gender into Grouping 12 Variable box and then click Define Groups and state which woman and man as group 1 and group 2 hit Continue We will need a 90 con dence interval in part c so go to Options to change it Group Statistics Std Error gender N Mean Std Deviation Mean score woman 18 14106 26436 6231 man 20 12125 32852 7346 Independent SamplesTe Leve ne s Te St for Equaiit of Variances Heat for Equaiit ofMeans 90 Con dence interval oftne Mean Std Error Difference F 810 t df Siq Hailed Difference Difference Lower UDDer score Equal variances assumed 862 359 2032 36 050 19806 9745 3353 36258 Equal variances not assumed 2056 35 587 047 19806 9633 3538 36073 What do we do with this Equal variances assumed and Equal variances not assumed Always go with the bottom row Equal variances not assumed This is the more conservative approach b Most studies have found that the population mean SSHA score for men is lower than the population mean score in a comparable group of women Test this supposition here c Give a 90 con dence interval for the difference in population means of SSHA scores of male and female firstyear students at this college 13 To summarize Chapters 6 and 7 Z vs t Z if you know the population standard deviation t if you know only the sample standard deviation This is usually the reallife situation and we will assume that we have only the sample standard deviation unless we are explicitly told otherwise Matched pairs vs 2sample comparison of means Matched pairs if all units are measured twice and or receive both treatments over time Before vs after is the most common example 2sample comparison of means if you have two separate groups but each unit is only measured once Men vs women is the most common example 14 n wn ruuvumwnm m pl a upm mm 1 mm J human49me m a sum 1 mumm w 4 we lulu3 w mm anquot u Mmmlmm I w

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