Question

Two proposed computer mouse designs were compared by recording wrist extension in degrees for 24 people who each used both mouse designs.† The difference in wrist extension was calculated by subtracting extension for mouse type B from the wrist extension for mouse type A for each person. The mean difference was reported to be 8.82 degrees. Assume that this sample of 24 people is representative of the population of computer users.

(a)

Suppose that the standard deviation of the differences was 14 degrees. Is there convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B? Test the relevant hypotheses using a 0.05 significance level. (Use

μ_d = μ_A − μ_B.)

State the appropriate null and alternative hypotheses.

H₀: μ_d < 0

H_a: μ_d > 0

H₀: μ_d > 0

H_a: μ_d < 0

    

H₀: μ_d = 0

H_a: μ_d ≠ 0

H₀: μ_d = 0

H_a: μ_d < 0

H₀: μ_d = 0

H_a: μ_d > 0

Find the test statistic and P-value. (Use a table or SALT. Round your test statistic to one decimal place and your P-value to three decimal places.)

t = P-value =

State the conclusion in the problem context.

We reject H₀. There is convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B. We fail to reject H₀. There is not convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B.     We fail to reject H₀. There is convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B. We reject H₀. There is not convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B.

(b)

Suppose that the standard deviation of the differences was 38 degrees. Is there convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B? Test the relevant hypotheses using a 0.05 significance level. (Use

μ_d = μ_A − μ_B.)

State the appropriate null and alternative hypotheses.

H₀: μ_d < 0

H_a: μ_d > 0

H₀: μ_d > 0

H_a: μ_d < 0

    

H₀: μ_d = 0

H_a: μ_d ≠ 0

H₀: μ_d = 0

H_a: μ_d < 0

H₀: μ_d = 0

H_a: μ_d > 0

Find the test statistic and P-value. (Use a table or SALT. Round your test statistic to one decimal place and your P-value to three decimal places.)

t = P-value =

State the conclusion in the problem context.

We reject H₀. There is convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B. We fail to reject H₀. There is not convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B.     We fail to reject H₀. There is convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B. We reject H₀. There is not convincing evidence that the mean wrist extension for mouse type A is greater than for mouse type B.

(c)

Briefly explain why different conclusions were reached in the hypothesis tests of parts (a) and (b).

The larger t test statistic in part (a) indicates that it is more likely that, assuming the population mean difference is zero, a difference of 8.82 degrees would be observed when the standard deviation of the differences was 14 than when it was 38. The smaller t test statistic in part (a) indicates that it is less likely that, assuming the population mean difference is zero, a difference of 8.82 degrees would be observed when the standard deviation of the differences was 14 than when it was 38.     The larger t test statistic in part (a) indicates that it is less likely that, assuming the population mean difference is zero, a difference of 8.82 degrees would be observed when the standard deviation of the differences was 14 than when it was 38. The smaller t test statistic in part (a) indicates that it is more likely that, assuming the population mean difference is zero, a difference of 8.82 degrees would be observed when the standard deviation of the differences was 14 than when it was 38.

Answer 1

Solution: Given na 24, od 8.82 a) Hypothesis Ho: Mdo Ha: Ma> o Given Std den SE lly d-Ma 8.82-0 Test statistic 14/624 7 3.086 t 3.1 df = - = 23, t=3.1 P-value for one-tailed fest (right tail) is p=0.002609 P-vana 0.003 Here p-value ed=0.05. Hence we reject to Conclusion: we reject Ho. There is convincing evidence that the weist extension for mouse type A is greater than for mouse type B. meam 5) sad den 38. Test Statistic 1.137 38/524 P-value 0.1336 p-value & 0.134 Here p-value > dno.05 we fail Conclusion : we fail to reject to. There that the mean wrist extension for mouse u to reject to not convincing evidence is greater than for type A mouse Bo

c)  The larger t test statistic in part (a) indicates that it is less likely that, assuming the population mean difference is zero, a difference of 8.82 degrees would be observed when the standard deviation of the differences was 14 than when it was 38.