Question

A drug manufacturer uses two production facilities to produce a pain reliever. The amount of the active ingredient of the drug in the capsules at the two​ facilities, X₁ and X₂, are normally distributed random variables. The desire of the quality control manager is that the population mean amounts of the active ingredient in the​ capsules, μ₁ and μ₂​, be equal. Recent tests on small samples have indicated a noticeable increase in the amount of the active ingredient in capsules coming from Plant​ #1. The manager decides to select larger samples from each plant and test the hypotheses H₀: μ₁−μ₂ ≤ 0 and H_A​: μ₁−μ₂ ​> 0. The results from the 2 samples are given below. The manager is not willing to assume that the variances in the two groups are equal. Based on these​ results, which of the following is​ true?

X₁ = ​52.1; X₂ =​ 49.9; S₁ = ​2.3; S₂ = ​1.9; n₁ = ​40; n₂ = ​37; df = 74

A. If the null hypothesis is not rejected for α = ​.005, a Type II error has occurred.

B. For a level of significance of α = ​.01, the difference in the sample means is statistically significant.

C. Using a level of significance of α = ​.01, the null hypothesis should not be rejected.

D. For a level of significance of α = ​.005, a Type I error will be made if the null hypothesis is false.

E. The​ p-value for the test statistic is greater than .005.

Answer 1

Test Statistic :-
t = (X̅1 - X̅2) / SP √ ( ( 1 / n1) + (1 / n2))



t = ( 52.1 - 49.9) / 2.1175 √ ( ( 1 / 40) + (1 / 37 ) )
t = 4.555

Test Criteria :-
Reject null hypothesis if t > t(α, n1 + n2 - 2)
Critical value t(α, n1 + n1 - 2) = t( 0.005 , 40 + 37 - 2) = 2.643
t > t(α, n1 + n2 - 2) = 4.555 > 2.643
Result :- Reject Null Hypothesis

Decision based on P value
P - value = P ( t > 4.555 ) = 0
Reject null hypothesis if P value < α = 0.005 level of significance
P - value = 0 < 0.005 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

For α = 0.01

Test Criteria :-
Reject null hypothesis if t > t(α, n1 + n2 - 2)
Critical value t(α, n1 + n1 - 2) = t( 0.01 , 40 + 37 - 2) = 2.377
t > t(α, n1 + n2 - 2) = 4.555 > 2.377
Result :- Reject Null Hypothesis

Decision based on P value
P - value = P ( t > 4.555 ) = 0
Reject null hypothesis if P value < α = 0.01 level of significance
P - value = 0 < 0.01 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

B. For a level of significance of α = ​.01, the difference in the sample means is statistically significant.

D. For a level of significance of α = ​.005, a Type I error will be made if the null hypothesis is false.