Question

A survey of 1000 teenage music fans from the United Kingdom indicated around $80\%$ of them are listening to streamed music on their computers and mobile devices every day. Suppose you decide to interview a random sample of 150 U.S. teenage music fans. Assume for now that their behavior is similar to the U.K. teenagers. (a) What is the distribution of the number of U.S. teenagers, $X$, in the sample that listen to streamed music daily. Name the distribution and indicate the values of all parameters. (b) Calculate the mean and standard deviation of the distribution found in part (a). (c) What is the probability that at least 132 of the 150 listen to streamed music daily? You may use the distribution function in

associated with your answer to part (a) to calculate this probability. (d) Use the Normal approximation to find the probability in part (c) using the relevant distribution function in
. (e) Is the value in part (d) a decent approximation of the true probability calculated in part(c)? Why or why not? (f) Calculate the mean and standard deviation of the distribution of the sample proportion, $\hat{p}$, that listen to streamed music daily. (g) What is the (approximate) probability that the sample proportion exceeds $88\%$? You may use the appropriate distribution function in
to obtain your answer. (h) How do the means and standard deviations in questions 1f and 2f compare?

Answer 1

The distribution of the number of U.S. teenagers that listen to streamed music daily follows a binomial distribution with parameters n = 150 and p = 0.8. The mean is 120 and the standard deviation is 4.9. The probability that at least 132 of the 150 U.S. teenagers listen to streamed music daily can be calculated using the binomial distribution function.

Step-by-step explanation:

a) The distribution of the number of U.S. teenagers, X, in the sample that listen to streamed music daily follows a binomial distribution. The parameters of this distribution are n = 150 (sample size) and p = 0.8 (probability of listening to streamed music daily based on the UK survey).

b) To find the mean and standard deviation of this distribution, we use the formulas:

mean = n * p = 150 * 0.8 = 120,

standard deviation = sqrt(n * p * (1-p)) = sqrt(150 * 0.8 * (1-0.8)) = 4.9.

c) To calculate the probability that at least 132 of the 150 U.S. teenagers listen to streamed music daily, we can use the binomial distribution function. The probability can be calculated as P(X >= 132) = 1 - P(X < 132), where X follows a binomial distribution with parameters n = 150 and p = 0.8.