Question

A company that sells dishware has two plants. In a quality control inspection of a random sample of 200 dishes from plant A, 8% of the dishes had at least one defect. In a random sample of 200 dishes from plant B, 5% of the dishes had at least one defect. To determine if there is convincing evidence that the true proportion of defective dishes from plant A is more than the true proportion of defective dishes from plant B, we test the hypotheses H0: pA - pB = 0 versus Ha: pA - pB > 0 and obtain a p-value of 0.112.

(a) Which of the following is an appropriate interpretation of this p-value?

(a)If the true proportion of defective dishes at plant A is more than the true proportion of defective dishes at plant B, there is a 0.112 probability of getting samples in which the difference p̂_A - p̂_B is equal to 0.03.

(b)If the true proportions of defective dishes at the two plants are equal, there is a 0.112 probability of getting samples in which the difference p̂_A - p̂_B is equal to 0.03.    

(c)The probability of making a Type I error is 0.112.

(d)The probability that the true proportion of defective dishes at plant A is more than the true proportion of defective dishes at plant B is 0.112.

(e) If the true proportions of defective dishes at the two plants are equal, there is a .112 probability of getting samples in which the difference p̂_A - p̂_B is greater than or equal to 0.03.

Suppose the company adds a third plant (plant C), and in a quality control inspection of a random sample of 200 dishes from plant C, 3% of the dishes have a defect. The company tests whether plant A has a higher proportion of defective dishes than plant C using H_a: p_A - p_C > 0. Compared to the previous test (A-B):
the new test of A-C would have  ---[ the same ,an indeterminate, a larger, a smaller test statistic] and  --- [a larger, a smaller, an indeterminate,the same p-value.]

Answer 1

Option: (e) If the true proportions of defective dishes at the two plants are equal, there is a .112 probability of getting samples in which the difference p̂A - p̂B is greater than or equal to 0.03.

Minitab output:

Test and CI for Two Proportions

Sample X N Sample p
A 16 200 0.080000
B 10 200 0.050000

Difference = p (A) - p (B)
Estimate for difference: 0.03
95% lower bound for difference: -0.0104748
Test for difference = 0 (vs > 0): Z = 1.22 P-Value = 0.112

Test and CI for Two Proportions

Sample X N Sample p
A 16 200 0.080000
C 6 200 0.030000

Difference = p (A) - p (C)
Estimate for difference: 0.05
95% lower bound for difference: 0.0127267
Test for difference = 0 (vs > 0): Z = 2.19 P-Value = 0.014

Answer: The new test of A-C would have a larger test statistic (since value of test statistic for A-C=2.19 and  value of test statistic for A-B=1.22) and a smaller p-value (since p-value for A-C=0.014, and p-value for A-B=0.112) .