Sampling Distribution of the Sample Proportion — Quiz
Question 5 • Math 4
Movie critics claim that 68 percent of adults and 79 percent of teenagers would recommend seeing the newest action movie. Suppose a random sample of 43 adults and 52 teenagers is selected. Let hat p sub cap A and hat p sub cap t be the sample proportions of adult and teenage moviegoers, respectively, who would recommend this movie. Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of hat p sub cap A minus hat p sub cap t ?
Answer
A
The difference (adult – teenager) in the sample proportions of those who would recommend this action movie varies about 0 point 0 4 3 from the true difference in proportions.
B
The difference (adult – teenager) in the sample proportions of those who would recommend this action movie varies about 0 point 1 7 6 from the true difference in proportions.
C
The difference (adult – teenager) in the sample proportions of those who would recommend this action movie varies about 0 point 0 0 8 from the true difference in proportions.
D
The difference (adult – teenager) in the sample proportions of those who would recommend this action movie varies about 0 point 0 9 1 from the true difference in proportions.
Explanation
For the difference between two independent sample proportions, the standard deviation is √[pA(1 − pA)/nA + pt(1 − pt)/nt]. Substituting the values gives √[(0.68 × 0.32)/43 + (0.79 × 0.21)/52] ≈ √0.00825 ≈ 0.091. Thus, the adult-minus-teenager sample-proportion difference typically varies about 0.091 from the true difference, which is choice D.
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