Exam 10: Two-Sample Tests

SCENARIO 10-9 The following EXCEL output contains the results of a test to determine whether the proportions of satisfied customers at two resorts are the same or different. Hypothesized Difference 0 Level of Significance 0.05 Group 1 Number of Items of Interest 160 Sample Size Group 2 Number of Items of Interest 172 Sample Size Intermediate Calculations Group 1 Proportion 0.8 Group 2 Proportion 0.688 Difference in Two Proportions 0.112 Average Proportion 0.737777778 Z Test Statistic 2.684103363 Two-Tail Test Lower Critical Value -1.959963985 Upper Critical Value 1.959963985 2-tailed p-Value 0.007272462 -Referring to Scenario 10-9, if you want to test the claim that "Resort 1 (Group 1) has a lowerproportion of satisfied customers compared to Resort 2 (Group 2)", you will use

SCENARIO 10-13 The amount of time required to reach a customer service representative has a huge impact on customer satisfaction.Below is the Excel output from a study to see whether there is evidence of a difference in the mean amounts of time required to reach a customer service representative between two hotels.Assume that the population variances in the amount of time for the two hotels are not equal. t-Test: Two-Sample Assuming Unequal Variances Hotel 1 Hotel 2 Mean 2.214 2.0115 Variance 2.951657 3.57855 Observations 20 20 Hypothesized Mean Difference 0 df 38 t Stat 0.354386 P (T<=t) one-tail 0.362504 t Critical one-tail 1.685953 P ( T < t) two-tail 0.725009 t Critical two-tail 2.024394 -Referring to Scenario 10-13, suppose $\alpha$ = 0.05.Which of the following represents the correct conclusion?

SCENARIO 10-12 A quality control engineer is in charge of the manufacture of USB flash drives.Two different processes can be used to manufacture the flash drives.He suspects that the Kohler method produces a greater proportion of defects than the Russell method.He samples 150 of the Kohler and 200 of the Russell flash drives and finds that 27 and 18 of them, respectively, are defective.If Kohler is designated as "Group 1" and Russell is designated as "Group 2," perform the appropriate test at a level of significance of 0.01. -Referring to Scenario 10-12, the null hypothesis should be rejected.

SCENARIO 10-6 To investigate the efficacy of a diet, a random sample of 16 male patients is selected from a population of adult males using the diet.The weight of each individual in the sample is taken at the start of the diet and at a medical follow-up 4 weeks later.Assuming that the population of differences in weight before versus after the diet follow a normal distribution, the t-test for related samples can be used to determine if there was a significant decrease in the mean weight during this period.Suppose the mean decrease in weights over all 16 subjects in the study is 3.0 pounds with the standard deviation of differences computed as 6.0 pounds. -Referring to Scenario 10-6, if we were interested in testing against the two-tail alternative that $\mu$ D is not equal to zero at the $\alpha$ = 0.05 level of significance, the null hypothesis would (be rejected/not be rejected).

SCENARIO 10-3 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles.She took 3 of each brand and determined their maximum downhill speeds.The results are presented in miles per hour in the table below. Trial Barth Tornado Reiser Shaw 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to SCENARIO 10-3, based on the Tukey-Kramer procedure with an overall level of significance of 0.05, the retailer would decide that there is no significant difference between any pair of mean speeds.

SCENARIO 10-6 To investigate the efficacy of a diet, a random sample of 16 male patients is selected from a population of adult males using the diet.The weight of each individual in the sample is taken at the start of the diet and at a medical follow-up 4 weeks later.Assuming that the population of differences in weight before versus after the diet follow a normal distribution, the t-test for related samples can be used to determine if there was a significant decrease in the mean weight during this period.Suppose the mean decrease in weights over all 16 subjects in the study is 3.0 pounds with the standard deviation of differences computed as 6.0 pounds. -Referring to Scenario 10-6, the critical value for a one-tail test of the null hypothesis of no difference at the $\alpha$ = 0.05 level of significance is .

SCENARIO 10-12 A quality control engineer is in charge of the manufacture of USB flash drives.Two different processes can be used to manufacture the flash drives.He suspects that the Kohler method produces a greater proportion of defects than the Russell method.He samples 150 of the Kohler and 200 of the Russell flash drives and finds that 27 and 18 of them, respectively, are defective.If Kohler is designated as "Group 1" and Russell is designated as "Group 2," perform the appropriate test at a level of significance of 0.01. -Referring to Scenario 10-12, the value of the test statistic is .

SCENARIO 10-2 A realtor wants to compare the mean sales-to-appraisal ratios of residential properties sold in four neighborhoods (A, B, C, and D).Four properties are randomly selected from each neighborhood and the ratios recorded for each, as shown below. A: 1.2,1.1,0.9,0.4 C: 1.0,1.5,1.1,1.3 B: 2.5,2.1,1.9,1.6 D: 0.8,1.3,1.1,0.7 Interpret the results of the analysis summarized in the following table: Source df SS MS F PR > F Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4.3644 -Referring to SCENARIO 10-2, the numerator and denominator degrees of freedom for Levene's test for homogeneity of variances at a 5% level of significance are, respectively,

SCENARIO 10-4 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds.She plants 15 fields, 5 with each variety.She then measures the crop yield in bushels per acre.Treating this as a completely randomized design, the results are presented in the table that follows. Trial Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to SCENARIO 10-4, the null hypothesis will be rejected at a level of significance of0.01 if the value of the test statistic is greater than .

SCENARIO 10-7 A buyer for a manufacturing plant suspects that his primary supplier of raw materials is overcharging.In order to determine if his suspicion is correct, he contacts a second supplier and asks for the prices on various identical materials.He wants to compare these prices with those of his primary supplier.The data collected is presented in the table below, with some summary statistics presented (all of these might not be necessary to answer the questions which follow).The buyer believes that the differences are normally distributed and will use this sample to perform an appropriate test at a level of significance of 0.01. Material Primary Supplier Secondary Supplier Difference 1 \ 55 \ 45 \ 10 2 \ 48 \ 47 \ 1 3 \ 31 \ 32 -\ 1 4 \ 83 \ 77 \ 6 5 \ 37 \ 37 \ 0 6 \ 55 \ 54 \ 1 Sum: \ 309 \ 292 \ 17 Sum of Squares: \ 17,573 \ 15,472 \ 139 -Referring to Scenario 10-7, the decision rule is to reject the null hypothesis if .

The Wall Street Journal recently ran an article indicating differences in perception of sexual harassment on the job between men and women.The article claimed that women perceived the problem to be much more prevalent than did men.One question asked to both men and women was: "Do you think sexual harassment is a major problem in the American workplace?" Some24% of the men compared to 62% of the women responded "Yes." Suppose that 150 women and200 men were interviewed.Construct a 90% confidence interval estimate of the difference between the proportion of women and men who think sexual harassment is a major problem in the American workplace.

SCENARIO 10-5 A hotel chain has identically small sized resorts in 5 locations in different small islands.The data that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5 locations. ROW Location A Location B Location C Location D Location E 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source df SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to SCENARIO 10-5, the null hypothesis for Levene's test for homogeneity of variances is a) $H _ { 0 } : \mu _ { A } = \mu _ { B } = \mu _ { C } = \mu _ { D }$ b) $H _ { 0 } : M _ { A } = M _ { B } = M _ { C } = M _ { D }$ c) $H _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 } = \sigma _ { C } ^ { 2 } = \sigma _ { D } ^ { 2 }$ d) $H _ { 0 } : \pi _ { A } = \pi _ { B } = \pi _ { C } = \pi _ { D }$

SCENARIO 10-3 A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis.Assume that the two population variances are equal.A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes. Gotham: $\bar { X } _ { \mathrm { G } } = 35 \text { months, } \quad S _ { \mathrm { G } } { } ^ { 2 } = 900 \quad \text { Metropolis: } \quad \bar { X } _ { \mathrm { M } } = 50 \text { months, } \mathrm { S } _ { \mathrm { M } } { } ^ { 2 } = 1050$ -Referring to Scenario 10-3, suppose $\alpha$ = 0.01.Which of the following represents the result of the relevant hypothesis test?

If you are comparing the mean sales among 3 different brands you are dealing with a three-way ANOVA design.

In a one-factor ANOVA analysis, the among sum of squares and within sum of squares must add up to the total sum of squares.

SCENARIO 10-3 A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis.Assume that the two population variances are equal.A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes. Gotham: $\bar { X } _ { \mathrm { G } } = 35 \text { months, } \quad S _ { \mathrm { G } } { } ^ { 2 } = 900 \quad \text { Metropolis: } \quad \bar { X } _ { \mathrm { M } } = 50 \text { months, } \mathrm { S } _ { \mathrm { M } } { } ^ { 2 } = 1050$ -Referring to Scenario 10-3, suppose $\alpha$ = 0.01.Which of the following represents the correct conclusion?

A statistics professor wanted to test whether the grades on a statistics test were the same for upper and lower classmen.The professor took a random sample of size 10 from each, conducted a test and found out that the variances were equal.For this situation, the professor should use a t test with independent samples.

SCENARIO 10-2 A researcher randomly sampled 30 graduates of an MBA program and recorded data concerning their starting salaries.Of primary interest to the researcher was the effect of gender on starting salaries. The result of the pooled-variance t-test of the mean salaries of the females (Population 1) and males (Population 2) in the sample is given below. Hypothesized Difference 0 Level of Significance 0.05 Population 1 Sample Sample Size 18 Sample Mean 99210 Sample Standard Deviation 13577 Population 2 Sample Sample Size 12 Sample Mean 105820 Sample Standard Deviation 11741 Difference in Sample Means -6610 t Test Statistic -1.37631 Lower-Tail Test Lower Critical Value -1.70113 p-Value 0.089816 -Referring to Scenario 10-2, the researcher was attempting to show statistically that the female MBA graduates have a significantly lower mean starting salary than the male MBA graduates.Which of the following is an appropriate alternative hypothesis? a) $H _ { 1 } : \mu _ { \text {females } } > \mu _ { \text {males } }$ b) $H _ { 1 } : \mu _ { \text {females } } < \mu _ { \text {males } }$ c) $H _ { 1 } : \mu _ { \text {femalea } } \neq \mu _ { \text {males } }$ d) $H _ { 1 } : \mu _ { \text {females } } = \mu _ { \text {males } }$

SCENARIO 10-3 A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis.Assume that the two population variances are equal.A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes. Gotham: $\bar { X } _ { \mathrm { G } } = 35 \text { months, } \quad S _ { \mathrm { G } } { } ^ { 2 } = 900 \quad \text { Metropolis: } \quad \bar { X } _ { \mathrm { M } } = 50 \text { months, } \mathrm { S } _ { \mathrm { M } } { } ^ { 2 } = 1050$ -Referring to Scenario 10-3, what is the test statistic for the difference between sample means?

SCENARIO 10-5 A hotel chain has identically small sized resorts in 5 locations in different small islands.The data that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5 locations. ROW Location A Location B Location C Location D Location E 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source df SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to SCENARIO 10-5, the total variation or SST is .