Question

Question:Paying for music downloads. If you use the Internet, have you ever paid to access or download music? This was one of the questions of interest in a Pew Internet & American Life Project Survey (October 2010). In a representative sample of 755 adults who use the Internet, 506 admitted

that they have paid to download music. Let p represent the true proportion of all Internet-using adults who have paid to download music.

a. Compute a point estimate of p.

b. Set up the null and alternative hypotheses for testing whether the true proportion of all Internet-using adults who have paid to download music exceeds.7.

c. Compute the test statistic for part b.

d. Find the rejection region for the test if α = 0.01.

e. Find the p-value for the test.

f. Make the appropriate conclusion using the rejection region.

g. Make the appropriate conclusion using the p-value.

Short answer

1. The point estimate of p is .
2. The null and alternative hypotheses are: H₀ : p= 0.7 Against H_a : p > 0.7 .
3. The value of the test statistic is z = -1.72
4. The rejection region is z > 2.33.
5. The p-value is 0.9573.
6. Do not reject the null hypothesis at a 1% significance level. There is no sufficient evidence to say that the true proportion of all Internet-using adults who have paid to download music exceeds 0.7.
7. Do not reject the null hypothesis. Therefore, there is not enough evidence to claim that the proportion of adults who paid to download music exceeds 0.7.

Step-by-step solution

Given Information

A random sample of 755 adults who use the Internet is taken. Out of these, 506 admitted that they have paid to download music.

Let p represent the true proportion of all Internet-using adults who have paid to download music.

The sample size is 755 is selected from the population. The sample proportion is 0.65.

Computing the point estimate of p

(a)

Definition of the point estimate -

It is the estimate of the population parameter in a single value. It represents the point on the number line hence called a point estimate.

The point estimate of the population proportion is the sample proportion .

The value of the sample proportion is:

The point estimate of p is .

Setting the null and alternative hypothesis

(b)

The alternative hypothesis generally represents the claim of the researcher. It is complementary to the null hypothesis.

A researcher wants to determine whether the true proportion exceeds 0.7.

The null and alternative hypotheses are: H₀ : p= 0.7 Against H_a : p > 0.7 .

Computing the test statistic 

(c)

The z-test is used to obtain the test statistic. It is the sampling distribution of sample proportion.

The test statistic is

The value of the test statistic is z = -1.72.

Obtaining the rejection region

(d)

The rejection region is obtained by using the critical value. For a right-tailed test, it lies on the right side of the critical value.

Referring to parts a. and b., the point estimate and test statistic are 0.70 and -1.72, respectively.

The z-critical value at a 1% level of significance for a right-tailed test is:

In the z-table, the z-score corresponding to the probability of 0.99 is the required value.

Therefore, the rejection region is z > 2.33.

Obtaining the p-value

(e)

Let p represents the true percentage of all adult Internet users who paid to download music.

The p-value is obtained by using the test statistic. For a right-tailed test, the p-value is the probability of getting the test statistic more than the calculated value of the test statistic.

The p-value for the right-tailed test is:

Hence the p-value is 0.9573.

Conclusion of the test using the rejection region

(f)

From the hypothesis test, the test statistic is z = -1.72, and the rejection region is z > 2.33.

From the rejection region, the decion regarding the null hypothesis is taken. The null hypothesis is not rejected if the test statistic is outside of the rejection region.

Here the test statistic does not lie in the rejection region. Do not reject the null hypothesis at a 1% significance level.

There is no sufficient evidence to say that the true proportion of all Internet-using adults who have paid to download music exceeds 0.7.

Conclusion of the test using the p-value

(g)

The p-value for the test is 0.9573, and the significance level is α = 0.01.

The p-value is used to take the decision about the hypothesis. The null hypothesis is rejected if the significance exceeds the p-value.

The p-value for the test is more significant than 0.05, hence do not reject the null hypothesis.

Therefore, there is not enough evidence to claim that the proportion of adults who paid to download music exceeds 0.7.