# OSU BUSMGT 2320 - Comparisons of Two Populations (mu) Additional Practice Solutions (9 pages)

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## Comparisons of Two Populations (mu) Additional Practice Solutions

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- Ohio State University
- Course:
- Busmgt 2320 - Decision Sciences: Statistical Techniques

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Comparisons Additional Practice Problems Solutions 1 If some natural relationship exists between each pair of observations that provides a logical reason to compare the first observation of sample 1 with the first observation of sample 2 the second observation of sample 1 with the second observation of sample 2 and so on the samples are referred to as a matched samples definition b independent samples c weighted samples d random samples 2 True or False A researcher is curious about the effect of sleep on students test performances He chooses 50 students and gives each two tests one given after four hours of sleep and one after eight hours of sleep The test the researcher should use would be matched pairs t test Each student is measured twice A student s general understanding of the subject matter analytical skills etc will remain the same for both tests 3 Consider the following hypothesis test Ho 1 2 1 7 n1 20 X 1 25 2 H1 1 2 1 7 n2 35 X 2 20 4 s1 4 0 05 s2 7 a What is the appropriate test statistic Z or t If t also indicate the degrees of freedom s22 49 Use the Satterthwaite Approximation Method since 2 3 0625 2 s1 16 x x 2 1 2 The test statistic is t obs 1 t 52 s12 s22 n1 n2 2 16 49 20 35 df 52 99 Use 52 df 2 2 2 2 2 2 16 49 s1 s2 n n 20 35 1 2 16 1 35 1 n1 1 n2 1 s12 s22 n 1 n2 2 Note If you do not have access to statistical computing software and the calculation of is not reasonable substitute the smaller of n1 1 and n2 1 for the df In this case you would substitute 19 This is far fewer df than the actual 52 so use if possible b What are the value s of the critical point s Express your answer to 3 decimal places The value will depend on access to computer or printed table If using printed table and df are not in table use next lower available df I m using df 40 t criticals t with 52 df row and 0 025 tail area column t criticals 2 021 c What is the 95 Confidence Interval estimate of 1 2 Express your answer to 3 decimal places X 1 X 2 t s12 s 22 16 49 25 2 20 4 2 021 1 802 7 798 20 35 n1 n2 d Reject Do not reject null hypotheses Reject H0 since the interval does not capture 1 7 e Conclusion There is statistically significant evidence at the 5 level that the difference between the means of population1 and population 2 does not equal 1 7 Since the 95 confidence interval contains values that are all 1 7 it is logical to conclude that the mean of population 1 mean of population 2 by more than 1 7 4 The number of degrees of freedom associated with the t test when the data are gathered from a matched pairs experiment with 10 pairs is a 10 b 20 c 9 There will be 10 differences One standard deviation will be calculated from the sample data based on those 10 differences The df nD 1 10 1 d 18 Use the following narrative to answer questions 5 and 6 Automobile insurance appraisers examine cars that have been involved in accidental collisions to assess the cost of repairs An insurance executive is concerned that different appraisers produce significantly different assessments In an experiment 10 cars that have recently been involved in accidents were shown to two appraisers Each assessed the estimated repair costs These results are shown below Car 1 2 3 4 5 6 7 8 9 10 Appraiser 1 1650 360 640 1010 890 750 440 1210 520 690 Appraiser 2 1400 380 600 920 930 650 410 1080 480 770 D 250 20 40 90 40 100 30 130 40 80 Sum 540 These calculations would usually be done with technology help xD 540 54 10 250 54 2 80 54 2 9 s D 94 7746 sD 5 What does D1 equal a 1650 360 the range of repair cost assessments for Appraiser 1 b 816 the average repair cost assessment for Appraiser 1 c 762 816 the difference in average repair cost assessments for Appraiser 1 and Appraiser 2 d 1650 1400 the difference in Appraiser 1 s repair cost assessment for car 1 and Appraiser 2 s repair cost assessment for car 1 The 2 appraisers cost assessments are matched to the car 6 Can the executive conclude at the 5 significance level that the appraisers differ in their assessments Solution H0 D 0 H1 D 0 difference in assessments no suggestion that one is higher than the other Test Statistic t obs xD 0 54 0 1 80 sD 94 7746 10 nD P value two sided test Using the t table with 9 df 2P t 1 80 is between 2 05 and 2 10 Using Minitab 0 05270 0 05270 2 0 025 2 0 025 1 8 2 262 0 1 8 t 9 t 2 262 Since P value 0 05 or 2 262 tobs 1 80 2 262 there is not statistically significant evidence of a difference on average in repair cost assessments made by the two appraisers Excel Data Analysis t Test Paired Two Sample for Means Appraiser 1 816 153560 10 0 982375856 Mean Variance Observations Pearson Correlation Hypothesized Mean Difference df t Stat P T t one tail t Critical one tail P T t two tail t Critical two tail Appraiser 2 762 105506 6667 10 0 9 1 801780416 0 05254704 1 833112933 0 10509408 2 262157163 StatTools Output Hypothesis Test Paired Sample Sample Size Sample Mean Sample Std Dev Hypothesized Mean Alternative Hypothesis Standard Error of Mean Degrees of Freedom t Test Statistic p Value Null Hypoth at 10 Significance Null Hypoth at 5 Significance Null Hypoth at 1 Significance Appraiser 1 Appraiser 2 10 54 94 77458637 0 0 29 97035572 9 1 8018 0 1051 Don t Reject Don t Reject Don t Reject Minitab Output Paired T Test and CI Appraiser 1 Appraiser 2 Paired T for Appraiser 1 Appraiser 2 Appraiser 1 Appraiser 2 Difference N 10 10 10 Mean 816 762 54 0 StDev 392 325 94 8 SE Mean 124 103 30 0 95 CI for mean difference 13 8 121 8 T Test of mean difference 0 vs not 0 T Value 1 80 P Value 0 105 7 Two types of new cars Type A and Type B are being considered for purchase by a fleet manager for a government agency An important criterion in the purchase decision is gas mileage The fleet manager has reason to think that Type A gets better gas mileage literature word of mouth etc but isn t convinced He is able to arrange a test of both types of cars A random sample of 36 cars of Type A had an average …

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