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# 547 Study Guide for STAT 30100 with Professor Zhao 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 In Chapter 6 we knew the population standard deviation 0 Con dence interval for the population mean u f i z Used the distribution f N N u 0X n f 05 Hypothesis test statistic for the population mean u Z0 O39 W In Chapter 7 we don t know the population standard deviation 039 Use the sample standard deviation s Con dence interval for the population mean u f i t i n Hypothesis test statistic for the population mean u to s J f 35 Sometimes you ll see the symbol for standard error 639 T n tn 1 Using the t dist bution 0 Suppose that an SRS of size n is drawn from aNJ 039 population 0 There is a different t distribution for each sample size so tk stands for the t distribution with k degrees of freedom 0 Degrees of freedom k n 7 1 sample size 7 l 0 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 0 Table shows upper tails only so 0 ift is negativePt lt t Pt gt lt l o if you have a 2sided test multiply the Pt gt lt by 2 to get the area in both tails o The normal table showed lower tails only so the ttable is backwards Finding t on the table Start at the bottom line to get the right column for your con dence 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 nfsnn dsnse mtgrmlfmmmamex An agneuuura1 expen perfums a study cu measureyxeld un alumatu eld Studying 1n plats ufland she ndsthe rnean yxeldxs 34 bushels vmh a sample standard dematmn uf 12 75 and a 95 cun dence interval fur the unknuwn nunulauun rnean yreld uftumatues Example afrssrfmnmnmss Cundun ahyputhesxs testwnh u as re deta mmexfthe rnean yreld uftumatues 151355 than 42 bushels State yuur cundusmn m terms quhe stury Alsu draw a picture ufthet curve wuh the number and symhnl fur the rnean yuu use m yuur nuu hyputhesxs m the sample mean 7 the standard errur Ex 7 f 1 n u and Lhatest statistic 1n Alsu shadethe zppmpnatepan quhe curvewhxch shuws the P39ValllE 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 a 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 Using this SPSS output OneSam ple Test Test Value 105 90 Confidence Interva of the Difference Mean t df Sig 2tailed Difference Lower Upper radon detector readings 319 11 755 8667 5739 4005 what would your tcurve with shaded PValue look like if you had hypotheses of H0u105 Ha u 105 H0u105 Ha ult105 H0u105 Ha ugt105 You must choose your hypotheses BEFORE you examine the data When in doubt do a twosided test 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 80 83 in your book will help you see the general shape of your data 0 t procedures are quite robust against nonnormality of the population except in the case of outliers or strong skewness 0 larger samples 7 improve the accuracy of the t distribution Some guidelines for inference on a single mean 0 n lt 15 Use t procedures if data close to normal If data nonnormal or if outliers are present do not use t o 15ltn 540 Use t procedures except in the presence of outliers or strong skewness o 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 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 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 If the skewness is extreme or if the 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 your book What do you do when you have 2 lists of data instead of 1 First decide whether you have 1 sample with 2 measurements OR 2 independent samples with one measurement each l Matched Pairs covered in 71 0 One group of individuals with 2 different measurements on them 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 67 and sample standard deviation s of these differences 3 Use the t distribution because the standard deviation is estimated from the data s 0 Confidence interval d it J o Hypothesis testing Hg udiff 0 t test statistic t d 0 0 sJ Example of Matched Pairs Exercise 731 Researchers are interested in whether Vitamin C is lost when wheat soy blend CSB is cooked as gruel Samples of gruel were collected and the Vitamin C content was measured in mg per 100 grams of gruel before and after cooking Here are the results Sample 1 2 3 4 5 Mean St DeV Before 73 79 86 88 78 808 614 After 20 27 29 36 17 258 753 Before 53 53 57 52 61 55 394 After a Set up an appropriate hypothesis test for the population mean difference and carry it out for these data State your conclusions in a sentence b Find a 90 confidence interval for the population mean Vitamin C content loss 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 along with the difference in the score Conduct a test to determine whether the training could on average improve the quality of nursing provided in the population individuals Pretraining Post score training 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 10 Enter the pre and post training scores to SPSS Then Analyze 9Compare Means 9Paireal 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 Posttraining score 63212 8 182086 64377 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 pretra39rl39rlg 39 1 05125 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 Pair Posttraining score 1 PrHraining score 105125 147417 5212 18119 228369 2017 084 What s different What s the same Which one matches the way that you de ned Wm 11 2 2 Sample Comparison of Means covered in 72 o A group of individuals is divided into 2 different experimental groups 0 Each group has different individuals who may receive different treatments 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 uA 3 same as Hg uA ug 0 Ha uA gt 3 or Ha uA lt 3 or Ha uA B pick one o 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 f f t0 M N t d1str1but10n w1th df mm 71A 1 n8 l i n n A B 0 Con dence Interval for uA ug 2 2 3 S EA f8it ii where t t d1str1but10n w1th df m1n nA lnB l quotA quotB Equal sample sizes are recommended but not required l2 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 df 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 firstyear 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 ca1ry 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 13 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 Samples Test Levene s Test for Equaiit ofVariances Heat fo Equaiit of Means 90 Con dence intervai oftne Mean std Error Difference F Sig t df Sig 2etaiied Difference Difference Lower Upper score Equal variances assumed 862 359 2 032 36 050 19 806 9 745 3 353 36 258 Equal variances notassumed 2 056 35 587 047 19 806 9 633 3 538 36 073 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 Give a 90 con dence interval for the difference in population means of SSHA scores of male and female firstyear students at this college c 14 To summarize Chapters 6 and 7 Z if you know the population standard deviation t if you only know the sample standard deviation Matched pairs vs 2 sample comparison of means Matched pairs if all subjects are in one group and receive both treatments Two sample comparison of means if you have 2 distinct groups of subjects Other notes If you have a small sample be careful If you don t have enough observations to do boxplots or normal quantile plots you might have trouble looking for outliers too If the effect is large you can still probably see it but you might miss small effects For the 2samples degrees of freedom in a t test we are taking df min nA l nBl There is also a software approximation for the degrees of freedom that does a better job than the minimum way The pooled 2sample tprocedures won t be covered now but if you need to compare more than 2 groups you would need it Chapter 12 15 9x WWW WWW w uddn wn rnuvumwnm m pl a 39JPnIJA mm 1 mm J IHWWN P39I39NW n m cm a mm m mp Ignrngnwnw in x m m u Munulmm quot

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