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Homework

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides a homework of F Distribution and One-Way ANOVA as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Directions:

Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in the Table of Contents 14. Appendix.

Exercise 1

Three students, Linda, Tuan, and Javier, are given 5 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. Using a significance level of 10%, test the hypothesis that the three formulas produce the same mean weight gain.

Table 1: Weights of Student Lab Rats
Linda's rats Tuan's rats Javier's rats
43.5 47.0 51.2
39.4 40.5 40.9
41.3 38.9 37.9
46.0 46.3 45.0
38.2 44.2 48.6

Solution

  • a. HoHo size 12{H rSub { size 8{o} } } {}: μL=μT=μJμL=μT=μJ size 12{μ rSub { size 8{L} } =μ rSub { size 8{T} } =μ rSub { size 8{J} } } {}
  • c. dfn=2dfn=2 size 12{ ital "df" left (n right )=2} {}; dfd=12dfd=12 size 12{ ital "df" left (d right )="12"} {}
  • e. 0.67
  • f. 0.5305
  • h. Decision: Do not reject null; Conclusion: There is insufficient evidence to conclude that the means are different.

Exercise 2

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 below. Using a 5% significance level, test the hypothesis that the 3 mean commuting mileages are the same.

Table 2
working-class professional (middle incomes) professional (wealthy)
17.8 16.5 8.5
26.7 17.4 6.3
49.4 22.0 4.6
9.4 7.4 12.6
65.4 9.4 11.0
47.1 2.1 28.6
19.5 6.4 15.4
51.2 13.9 9.3

Exercise 3

Refer to Exercise 13.8.1. Determine whether or not the variance in weight gain is statistically the same among Javier’s and Linda’s rats.

Solution

  • c. dfn=4dfn=4 size 12{ ital "df" left (n right )=4} {}; dfd=4dfd=4 size 12{ ital "df" left (d right )=4} {}
  • e. 3.00
  • f. 2 0 . 1563 = 0 . 3126 2 0 . 1563 = 0 . 3126 size 12{2 left (0 "." "1563" right )=0 "." "3126"} {} . Using the TI-83+/84+ function 2-SampFtest, you get the the test statistic as 2.9986 and p-value directly as 0.3127. If you input the lists in a different order, you get a test statistic of 0.3335 but the p-value is the same because this is a two-tailed test.
  • h. Decision: Do not reject null; Conclusion: There is insufficient evidence to conclude that the variances are different.

Exercise 4

Refer to Exercise 13.8.2 above. Determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups.

For the next two problems, refer to the data from Terri Vogel’s Log Book.
http://cnx.org/content/m17132/latest/?collection=col10522/latest/

Exercise 5

Examine the 7 practice laps. Determine whether the mean lap time is statistically the same for the 7 practice laps, or if there is at least one lap that has a different mean time from the others.

Solution

  • c. dfn=6dfn=6 size 12{ ital "df" left (n right )=6} {}; dfd=98dfd=98 size 12{ ital "df" left (d right )="98"} {}
  • e. 1.69
  • f. 0.1319
  • h. Decision: Do not reject null; Conclusion: There is insufficient evidence to conclude that the mean lap times are different.

Exercise 6

Examine practice laps 3 and 4. Determine whether or not the variance in lap time is statistically the same for those practice laps.

For the next four problems, refer to the following data.

The following table lists the number of pages in four different types of magazines.

Table 3
home decorating news health computer
172 87 82 104
286 94 153 136
163 123 87 98
205 106 103 207
197 101 96 146

Exercise 7

Using a significance level of 5%, test the hypothesis that the four magazine types have the same mean length.

Solution

  • a. HoHo size 12{H rSub { size 8{o} } } {}: μd=μn=μh=μcμd=μn=μh=μc size 12{μ rSub { size 8{d} } =μ rSub { size 8{n} } =μ rSub { size 8{h} } =μ rSub { size 8{c} } } {}
  • b. Alternate Hypothesis: At least one pair of means is different
  • c. dfn=3dfn=3 size 12{ ital "df" left (n right )=3} {}; dfd=16dfd=16 size 12{ ital "df" left (d right )="16"} {}
  • e. 8.69
  • f. 0.0012
  • h. Decision: Reject null; Conclusion: There is sufficient evidence to conclude that the mean lengths are different.

Exercise 8

Eliminate one magazine type that you now feel has a mean length different than the others. Redo the hypothesis test, testing that the remaining three means are statistically the same. Use a new solution sheet. Based on this test, are the mean lengths for the remaining three magazines statistically the same?

Exercise 9

Which two magazine types do you think have the same variance in length?

Exercise 10

Which two magazine types do you think have different variances in length?

Exercise 11

A researcher wants to know if the mean time (in minutes) that people watch their favorite news station are the same. Suppose that the table below shows the results of a study.

Table 4
CNN FOX Local
45 15 72
12 43 37
18 68 56
38 50 60
23 31 51
35 22  
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.

Solution

  • c: df(n) = 2; df(d) = 14
  • d: F2,14F2,14 size 12{H rSub { size 8{o} } } {}
  • e: 4.08
  • f: 0.0401
  • h:
  • ii: Reject the null hypothesis
  • iv: There is sufficient evidence to conclude that the mean times are different.

Exercise 12

Are the means for the final exams the same for all statistics class delivery types? The table below shows the scores on final exams from several randomly selected classes that used the different delivery types.

Table 5
Online Hybrid Face-to-Face
72 83 80
84 73 78
77 84 84
80 81 81
81   86
    79
    82
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.

Exercise 13

Are the mean number of times a month a person eats out same for whites, blacks, Hispanics and Asians? Suppose that the table below shows the results of a study.

Table 6
White Black Hispanic Asian
6 4 7 8
8 1 3 3
2 5 5 5
4 2 4 1
6   6 7
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.

Solution

  • c: df(n) = 3; df(d) = 15
  • d: F3,15F3,15 size 12{H rSub { size 8{o} } } {}
  • e: 0.8853
  • f: 0.4711
  • h:
  • ii: Do not reject the null hypothesis
  • iv: There is insufficient evidence to conclude that the mean number of times are different.

Exercise 14

Are the mean number of daily visitors to a ski resort the same for the three types of snow conditions? Suppose that the table below shows the results of a study.

Table 7
Powder Machine Made Hard Packed
1210 2107 2846
1080 1149 1638
1537 862 2019
941 1870 1178
  1528 2233
  1382  
Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05.

Exercise 15

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 that the table below shows the results of a study.

Table 8
Saturday Sunday
75 44
62 137
18 58
0 82
150 61
124 39
94 19
50 127
62 99
31 141
73 60
118 73
  89
Assume that both distributions are normal. Use a level of significance of 0.05.

Solution

  • c: df(n) = 11; df(d) = 12
  • d: F11,12F11,12 size 12{H rSub { size 8{o} } } {}
  • e: 1.35
  • f: 0.6090
  • h:
  • ii: Do not reject the null hypothesis
  • iv: There is insufficient evidence to conclude that the variances are different.

Exercise 16

Are the variances for incomes on the East Coast and the West Coast the same? Suppose that the table below shows the results of a study. Income is shown in thousands of dollars.

Table 9
East West
38 71
47 126
30 42
82 51
75 44
52 90
115 88
67  
Assume that both distributions are normal. Use a level of significance of 0.05.

**Exercises 11 - 16 were contributed by Dr. Larry Green

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