Question

If you use the Internet, haveyou ever paid to access or download music? This was oneof the questions of interest in a recent Pew Internet and American Life Project Survey (October ). Telephoneinterviews were conducted on a representative sample of adults living in the United States. For this sample, adults admitted that they have paid to download music.

a.Use the survey information to find a point estimatefor the true proportion of U.S. adults who have paid todownload music.

b.Find an interval estimate for the proportion, part a.Useconfidence interval.

c.Give a practical interpretation of the interval,part b.Your answer should begin with “We areconfident. . ..”

d.Explain the meaning of the phrase “confident.”

e.How many more adults need to be sampled to reducethe margin of error in the confidence interval by half?

Short answer

a. The true proportion of point estimate is

$CI=\overline{X}\pm z\times \frac{\sigma }{\sqrt{n}}$

b. For confidence interval, the interval estimate is given by

$CI=\overline{X}\pm 1.645\times \frac{\sigma }{\sqrt{n}}$

c. It is applied for downloading music.

d. The meaning of$90\%$confidence interval is that it is interpreted for paid to download music users.

e. So, there are $90\%$adults need to be sampled to reduce the margin of error

Step-by-step solution

Given information

Given that telephone interviews were conducted on a representative sample of adults living in the United States. For this sample,adults admitted that they have paid to download music.

(a) Find the point estimate of the true proportion

The true proportion of the point estimate is given by

$CI=\overline{X}\pm z\times \frac{\sigma }{\sqrt{n}}$

(b) Find an interval estimate of the confidence interval

Forconfidence interval, the interval is given by

$CI=\overline{X}\pm 1.645\times \frac{\sigma }{\sqrt{n}}$

Here the z score is$z=1.645$

(c) Write the practical interpretation of the confidence interval

For$90\%$confidence interval, the interval is given by

$CI=\overline{X}\pm 1.645\times \frac{\sigma }{\sqrt{n}}$

Here the Z score is$z=1.645$

Here the total sample size $n=1003$ and the mean is given by

$\begin{array}{rcl}\overline{X}& =& \frac{{\sum }_{i=1}^n{X}_i}{n}\\ & =& \frac{506}{1003}\\ & =& 0.504\\ & & \end{array}$

And the variance is unknown. So here we have to find the variance. Then it is given by

$\begin{array}{rcl}0.90& =& 0.504\pm 1.645\times \frac{\sigma }{\sqrt{1003}}\\ 28.503& =& 15.96+1.645\sigma \\ \sigma & =& \frac{12.543}{1.645}\\ \sigma & =& 7.624\\ & & \end{array}$

So, here in this problem, to interpret that for confidence interval, it is applied for paid to download music.

(d) Explain the meaning of confidence interval

The meaning of confidence interval is that it is interpreted for paid to download music users.

(e) Find how many adults need to be sampled to reduce the margin of error in the confidence interval by half

For $90\%$confidence interval, the interval estimate is given by

$\begin{array}{rcl}CI& =& \overline{X}\pm 1.645\times \frac{\sigma }{\sqrt{n}}\\ & =& 0.504\pm 1.645\times \frac{7.624}{31.67}\\ & =& 0.504+0.394\\ & =& 0.907\\ & & \end{array}$

And

$\begin{array}{rcl}CI& =& 0.504-0.394\\ & =& 0.11\\ & & \end{array}$

So, there are$90\%$adults need to be sampled to reduce the margin of error.