Question

Suppose a recent poll shows that 40% of Americans drink soda every day. Amy, a researcher at a local university, takes a

random sample of 22 Americans. Let X represent the number of Americans in Amy's sample who drink soda every day.
Let and o represent the parameters of a normal distribution and let n and p represent the parameters of a binomial
distribution.
The sampling distribution of X is
(a) exactly binomial with n 22 and p 0.4.
(b) exactly normal with p = 8.8 and o = 2.298.
(c) approximately normal with = 0.4 and = C). 104.
(d) almost exactly binomial with n 22 and p 0.4.

Answer 1

The correct distribution of the number of Americans who drink soda every day in Amy's sample (X) would be exactly binomial with n = 22 and p = 0.4, not a normal distribution.

Step-by-step explanation:

The sampling distribution of X, which represents the number of Americans in Amy's sample who drink soda every day, would be binomial distribution because the sample size is fixed, each individual either drinks soda daily or does not (two possible outcomes), and the probability of an individual drinking soda daily remains constant (p = 0.4).

Therefore, the correct answer is (a) exactly binomial with n = 22 and p = 0.4. Note that while a binomial distribution can be approximated to a normal distribution under certain conditions (large sample size and np and n(1-p) are both greater than 5), in this case, the sample size is not large enough to necessitate this approximation.

Hence option (c) which suggests an approximate normal distribution does not apply here, and neither does option (b), as it incorrectly suggests an exact normal distribution with impossible parameters (where p should represent a probability and should not be equal to 8.8).