WP.12.1 ADDITIONAL TEST OF TWO POPULATION VARIANCES PROBLEMS

[WP.12.1]

WHITE PAPER TOPIC: ADDITIONAL TEST OF TWO VARIANCES PROBLEMS

I.               ADDITIONAL EXERCISES

[Problem 1]

Professor Spitler has two different routes to drop off his daughter at the hockey rink.  One is the back roads and the other is the expressway.  Over the last month, he tracked the drive times for each route.  From a sample of ten days, he found the mean time driving the expressway was 20.5 minutes with a standard deviation of 3.5 minutes.  From a sample of nine days, he found that the mean time using the back roads was 20 minutes with a standard deviation of 4.5 minutes.  At the 0.10 significance level, is there a difference in variation for the drive time using the back roads verses the expressway?

[Problem 2]

Daily Research Inc. conducted a study of the watch times for a streaming video service.  They were primarily interested in the differences in watching habits between men and women. From a sample of 16 women, they found the average was 7.5 hours per week with a standard deviation of 2.45 hours.  From a sample of 16 men, they found the average was 9.5 hours per week with a standard deviation of 1.45 hours.  At the 0.01 significance level, is there more variation in the watch time of women than men?

[Problem 3]

An investor has classified her stocks into two main portfolios: a growth fund and an income fund.  She tracked a sample of 12 growth stocks and found the sample standard deviation to be 3.5% over the course of the fiscal year.  During the same year, she tracked 14 income stocks and found the standard deviation to be 2.8%.  Test at the 0.01 level whether or not the amount of variation is higher in the growth fund.

[Problem 4]

A sample size of eight is taken from Population A and the calculated standard deviation is 1.20.  A sample size of ten is taken from Population B and the calculated standard deviation is 0.85.  At the 0.02 significance level, is there difference in the variation of the populations?

[Problem 5]

A sample size of twelve is taken from Population A and the calculated standard deviation is 98.  A sample size of twelve is taken from Population B and the calculated standard deviation is 108.  At the 0.05 significance level, is there more variation in Population B compared to A?

[Problem 6]

Two coworkers commute to and from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 5% level.  (Adapted; A.Holmes; P.2-9; R.1.27)

[Problem 7]

Two students are interested in whether or not there is variation in their test scores for a math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks the variation in his scores is lower.  Test the claim at the 0.01 significance level. (Adapted; A.Holmes; P.10-13; R.1.27)

[Problem 8]

Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up 20 hills. The first cyclist has a variance of 16.2 minutes and the second cyclist has a variance of 32.1 minutes. The cyclists want to see if their variances are the same or different.  Test the claim at the 0.10 significance level. (Adapted; A.Holmes; P.14-16; R.1.27)

[Problem 9]

Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat’s weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded.

Determine whether or not the variance in weight gain is statistically the same among Javier’s and Linda’s rats. Test at a significance level of 10%.  (Adapted; A.Holmes; H.55; R.1.27)

[Problem 10]

A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily one-way commuting mileage. The results are as follows:

Determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups. Use a 10% significance level.  (Adapted; A.Holmes; H.56; R.1.27)

[Problem 11]

Is the variance for the amount of money, in dollars, that shoppers spend on Saturdays at the mall the same as the variance for the amount of money that shoppers spend on Sundays at the mall? Suppose the table included below shows the results of a study.  Use a 2% significance level.  (Adapted; A.Holmes; H.59; R.1.27)

[Problem 12]

Are the variances for incomes on the East Coast and the West Coast the same? Suppose the table included below shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of 0.02. (Adapted; A.Holmes; P.60; R.1.27)

[Solutions]

Problem 1: H_o: s²_br = s²_ex H_a: s²_br ¹ s²_ex; Reject H_o if f > 3.23; f_calc = 1.653; Do not reject H_o

Problem 2: H_o: s²_w £ s²_m H_a: s²_w > s²_m; Reject H_o if f > 3.52; f_calc = 2.855; Do not reject H_o

Problem 3: H_o: s²_g £ s²_i H_a: s²_g > s²_i; Reject H_o if f > 4.02; f_calc = 1.563; Do not reject H_o

Problem 4: H_o: s²_A = s²_B H_a: s²_A ¹ s²_B; Reject H_o if f > 5.61; f_calc = 1.99; Do not reject H_o

Problem 5: H_o: s²_B £ s²_A H_a: s²_B > s²_A; Reject H_o if f > 2.82; f_calc = 1.21; Do not reject H_o

Problem 6: H_o: s²₂ £ s²₁ H_a: s²₂ > s²₁; Reject H_o if f > 2.17; f_calc = 1.951; Do not reject H_o

Problem 7: H_o: s²₁ £ s²₂ H_a: s²₁ > s²₂; Reject H_o if f > 3.70; f_calc = 2.867; Do not reject H_o

Problem 8: H_o: s²₁ = s²₂ H_a: s²₁ ¹ s²₂; Reject H_o if f > 2.17; f_calc = 3.926; Reject H_o

Problem 9: H_o: s²_J = s²_L H_a: s²_J ¹ s²_L; Reject H_o if f > 6.39; f_calc = 2.998; Do not reject H_o

Problem 10: H_o: s²_WC = s²_PM H_a: s²_WC ¹ s²_PM; Reject H_o if f > 3.79; f_calc = 9.012; Reject H_o

Problem 11: H_o: s²_SAT = s²_SUN H_a: s²_SAT ¹ s²_SUN; Reject H_o if f > 4.22; f_calc = 1.354; Do not reject H_o

Problem 12: H_o: s²_W = s²_E H_a: s²_W ¹ s²_E; Reject H_o if f > 7.19; f_calc = 1.232; Do not reject H_o

REVISION INFORMATION

Date                          Author                    Note

11.28.16               Spitler                     Initial Release

11.28.16               n/a                             A.Holmes Adapted; Problems 6-12 add. 

 http://cnx.org/contents/b56bb9e9-5eb8-48ef-9939-88b1b12ce22f@27.12

11.30.16               Spitler                     Problems 1-5 Solutions add.

04.12.17               Spitler                     Problems 6-12 Solutions Update.

04.12.17               Spitler                     OER Commons Release