Sampling Distribution of the Sample Proportion — Quiz
Question 3 • Math 4
Beet farmers claim that 76 percent of vegetarians like beets, while only 41 percent of carnivores like beets. Suppose a random samples of 21 vegetarians and 31 carnivores are asked if they like beets. Let hat p sub v and hat p sub c be the sample proportions of vegetarians and carnivores, respectively, who like beets. Which of the following is the correct shape and justification of the sampling distribution of hat p sub v minus hat p sub c ?
Answer
A
not approximately Normal because the expected numbers of successes and failures for the vegetarian group are not both at least 10
B
not approximately Normal because the expected numbers of successes and failures for each sample are all at least 10
C
bimodal because one population proportion is centered at 0 point 7 6 , while the other is centered at 0 point 4 1
D
approximately Normal because the expected numbers of successes and failures for each sample are all at least 10
Explanation
For the vegetarian sample, the expected number who like beets is 21(0.76) = 15.96, and the expected number who do not is 21(0.24) = 5.04. Since 5.04 is less than 10, the success-failure condition for an approximately Normal sampling distribution is not met. Thus the distribution is not approximately Normal; choice A gives the correct reason.
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