Question

a streaming service introduced a new ad - supported subscription plan. in a survey of 221 subscribers, 24 said they believe the plan offers poor value for the price. does this suggest that a majority of subscribers do not hold this belief?
test appropriate hypotheses at level 0.05. (let p be the proportion of subscribers who do not believe the plan offers poor value.)
a. choose the appropriate null and alternative hypotheses.
a. ( h_0:p<0.50,h_a:p = 0.50 )
b. ( h_0:p>0.50,h_a:p = 0.50 )
c. ( h_0:p = 0.50,h_a:p
eq0.50 )
d. ( h_0:p = 0.50,h_a:p<0.50 )
e. ( h_0:p = 0.50,h_a:p>0.50 )
b. compute the test statistic and associated p - value. (round each value as instructed)
( z=) 1 (round your answer to two decimal places.)
( p - value=) 2 (round your answer to four decimal places.)
c. what can you conclude based on the results of the hypothesis test performed above?
a. we should not reject the null hypothesis, because there is insufficient evidence to suggest that more than 50 of all subscribers do not share the belief that the subscription plan offers a poor value.
b. we should reject the null hypothesis, because there is insufficient evidence to suggest that more than 50 of all subscribers do not share the belief that the subscription plan offers a poor value.
c. we should not reject the null hypothesis, because there is sufficient evidence to suggest that more than 50 of all subscribers do not share the belief that the subscription plan offers a poor value.
d. we should reject the null hypothesis, because there is sufficient evidence to suggest that more than 50 of all subscribers do not share the belief that the subscription plan offers a poor value.
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Explanation:

Step1: Determine the hypotheses

We are testing if a majority (more than 50%) of subscribers do not believe the plan offers good value. So, the null hypothesis \(H_0:p = 0.50\) and the alternative hypothesis \(H_1:p>0.50\) (Option A).

Step2: Calculate the sample proportion \(\hat{p}\)

The sample size \(n = 221\) and the number of subscribers who do not believe the plan offers good value \(x = 111\). The sample proportion \(\hat{p}=\frac{x}{n}=\frac{111}{221}\approx0.5023\).

Step3: Calculate the test - statistic \(z\)

The formula for the test - statistic in a one - sample proportion test is \(z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\), where \(p_0 = 0.50\), \(n = 221\), and \(\hat{p}\approx0.5023\).

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\(\sqrt{221}\approx14.87\), so \(z=\frac{0.0023\times14.87}{0.5}\approx0.0684\)

Step4: Calculate the P - value

Since this is a right - tailed test (\(H_1:p>0.50\)), the P - value is \(P(Z>z)\). Using the standard normal distribution table, \(P(Z > 0.0684)\approx0.4721\) (using the fact that for \(z = 0.07\), \(P(Z>0.07)=0.4721\) from the standard normal table).

Step5: Make a decision

The significance level \(\alpha=0.05\). Since the P - value (\(0.4721\))>\(\alpha(0.05)\), we fail to reject the null hypothesis.

Answer:

a. A. \(H_0:p < 0.50,H_1:p = 0.50\)
b. \(z\approx0.07\), \(P - value\approx0.47\)
c. B. We should not reject the null hypothesis, because there is insufficient evidence to suggest that more than 50% of all subscribers do not share the belief that the subscription plan offers a poor value.