Part 2. According to a recent poll, 69% of adults who use the Internet have paid
ID: 3351692 • Letter: P
Question
Part 2. According to a recent poll, 69% of adults who use the Internet have paid to download music. In a random sample of n=750 adults who use the Internet, let p represent the proportion who have paid to download music. Complete parts a through e below.
a. Find the mean of the sampling distribution of ^p. E(^p) = 0.69
b. Find the standard deviation of the sampling distribution. Standard deviation symbol ^p= 0.0169
c. What does the Central Limit Theorem say about the shape of the sampling distribution of ^p?
d. Compute the probability that ^p with is less than 0.770.77. p(^p<0.77)=
Explanation / Answer
p= propertion of adults who use the internet have paid to download music =69% = 0.69
q= 1-p = 0.31
n = number of adults use the internet.=750
let p^ be the sample propetion of adults use the internet have to paid download music.
Sample propetion is an unbiased estimator population propertion.
E(p^) =p and Var (p^) =pq/n
i) The mean of the sampling distribution is
E(p^) = p = 0.69
ii) The standard deviation of the sampling distribution is
S.D. (p^) =sqrt ( p*q/n) = sqrt( 0.69*31/750) = 0.01688
S.D. (p^) = 0.0169
iii) by central limit theorem
z= (p^ -E(p^))/S.D (p^) ~ N(0,1)
by central limit theorem the sampling distribution of p^ tends to standard normal distibution. The shape of the sampling distribution of p^ is bell shaped.
iv) Required probability = P (p^ < 0.77)
= P ( (p^-E(p^)/S.D.(p^)) < ((0.77-0.69)/0.0169))
= P (z < 4.7337) ( z ~ N(0,1))
P (p^ < 0.77) = 1