Exam 15: Nonparametric Statistics

A wild life biologist believes that the median length of the fish in a lake is 36 cm. A random sample of 14 fish yields the following lengths: 21 21 23 25 25 27 28 30 31 33 36 38 42 44 Test the biologist's hypothesis at $\alpha = 0.05$

Given n₁=19, n₂=30, S=401, and H₁: m₁ $\neq$ m₂ , find the P-value.

The owners of a coffee stand hypothesize that the median number of sales during the hour from 10:00 AM to 11:00 AM is 25. They tabulated the following random sample of the number of sales during the time period. 22 28 26 24 26 26 27 29 23 27 29 28 29 24 23 Use the $\alpha = 0.05$ level of significance and provide the following to provide the following information: a. State appropriate null and alternate hypotheses. b. Compute the test statistic. c. Find the critical value. d. State a conclusion.

A consumer advice web site tested a fuel additive. The distance that 12 cars could travel on five gallons of gasoline was recorded without and then with the additive. The results were as follows. Car 1 2 3 4 5 6 7 8 9 10 11 12 No Additive 105 139 185 108 187 134 170 161 145 115 190 121 With additive 115 157 191 108 197 139 183 168 159 115 193 127 a) Compute the test statistic b) Can you conclude that the median distance traveled with the additive difference from the median Distance traveled with the additive? Use the α = 0.05 level of significance.

The length of time that customers spent eating dinner was compared with and without music playing. The times, in minutes, were as follows. Without Music 55 67 59 55 43 76 45 60 93 61 63 51 With Music 79 87 63 79 48 89 97 80 107 43 100 73 104 84 70 a) Compute the P-value. b) Can you conclude that the median times eating dinner differ with and without music? Use the $\alpha$ =0.01 level of significance.

Fill in the blank with the appropriate word or phrase. When performing the Rank-sum test we reject the null hypothesis when ? .

The sign test is performed to test $H _ { 0 } : m = 32 \text { versus } H _ { 1 } : m < 32$ . There are 14 positive signs and 19 negative signs in a test involving 33 samples. What is the value of the test statistic?

A sample of 10 students took a class online and 12 students took an equivalent class in a traditional classroom. Both classes were given the same final exam three weeks after the end of the courses. The scores were as follows. Online 67 84 63 94 65 78 63 72 82 76 Traditional 74 70 87 90 95 73 96 68 90 64 92 80 Can you conclude that the median score for the online class is less than for the traditional class? Use the α = 0.05 level of significance. a) State the null and alternate hypotheses. b) Compute the value of the test statistic. c) Compute the P-value. d) State a conclusion

Find the critical value at the $\alpha = 0.05$ = 0.05 level for the following sample, for testing $H _ { 0 } : m = 60$ versus :m60 . 77 64 79 79 42 80 74 54 60 63 50 56

For the following data, compute the test statistic and the critical value, and determine whether to reject $H _ { 0 }$ at the $\alpha = 0.05$ level. Sample A 69 60 68 64 78 65 60 76 60 75 Sample B 59 41 68 59 84 65 46 84 48 61

Given n₁=16, n₂=21, S=249, and H₁: m₁<m₂ , find the P-value.

Ten subjects were weighed before and after a new diet. The results were as follows. Subject 1 2 3 4 5 6 7 8 9 10 Before 151 183 222 176 193 202 150 200 178 190 After 135 164 206 188 199 202 139 195 178 198 a) Compute the test statistic b) Can you conclude that the median weight differs before and after the diet? Use the α = 0.05 level Of significance.

The following data was collected as part of a study examining whether there is a difference between the number of hours men and women watch television. The values represent the number of hours a Subject watched television on a designated Tuesday night. In the process of computing the test Value the data from both samples should be combined, arranged in order, and ranked according to Each group. Calculate the sum of the ranks for both groups. Lower values rank ahead of higher Ones. Men 2.0 1.5 3.0 2.5 2.0 1.0 0.0 2.0 1.5 2.5 2.0 2.0 Women 2.0 2.5 1.0 1.0 1.5 2.5 2.0 1.0 2.0 1.5 1.0 0.0

The sign test is performed to test $H _ { 0 } : m = 45 \text { versus } H _ { 1 } : m \neq 45$ . There are 13 positive signs and 5 negative signs. What is the value of the test statistic?

Given $n _ { 1 } = 18 , n _ { 2 } = 27 , \mathrm {~S} = 398 , \text { and } H _ { 1 } : m _ { 1 } < m _ { 2 }$ compute $\sigma _ { S }$

Given n₁=20, n₂=26, S=516, and H₁: m₁>m₂ compute $Z$

Six second-graders tried tossing a ball into a basket ten times each. Their teacher then suggested a different way of tossing the ball, and the six students tried again. The number of successful tosses, Before and after the teacher's suggestion, are shown below. Student Before the suggestion (out of 10) 2 4 5 2 5 4 After the suggestion (out of 10) 4 4 2 4 2 4 Compute S, the test statistic for a signed-rank test.

The sign test is performed to test $H _ { 0 } : m = 400$ versus $H _ { 1 } : m < 400$ . There are 13 plus signs, 24minus signs, and 2 zeros. a. What is the test statistic? b. Is H₀ rejected at the $\alpha$ =0.05 level? c. Is H₀ rejected at the $\alpha$ =0.01 level?

Monthly rents were recorded for a sample of 36 apartments in a certain city. The results were as follows. 880 1,190 960 1,170 1,040 1,050 1,310 990 1,070 930 1,010 1,020 1,300 910 1,340 930 960 850 1,340 1,230 1,020 1,080 1,050 950 880 1,230 900 860 1,060 1,320 930 860 940 1,160 900 940 Can you conclude that the median rent is less than $1,100 per month? Use the $\alpha = 0.05$ level of significance. a. State appropriate null and alternate hypotheses. b. Compute the test statistic. c. Find the critical value. d. State a conclusion.

Heights, in feet, of a sample of 24 mature oak trees in a forest were measured. The results were as follows. 58 58 57 48 61 59 59 55 65 65 41 63 41 62 55 62 60 50 63 60 56 50 55 57 Can you conclude that the median height of oak trees in this forest is greater than 55 feet? Use the α = 0.05 level of significance. a. State appropriate null and alternate hypotheses. b. Compute the test statistic. c. Find the critical value. d. State a conclusion.