Question

In a survey conducted a few years ago 41% of Americans said they wanted Internet access on their cell phones. Suppose AT&T believes this percentage has increased and would like to verify their hypothesis. In a random sample of 100 Americans, 52 indicated that they wanted Internet access on their cell phones. Using a 0.05 level of significance, answer the questions below. a. Set up the null and alternative hypothesis for this problem. b. Which distribution (normal or t) should be used for this problem? WHY? c. Prove the above distribution cited can be used in this problem. d. State the level of significance. e. Create a decision rule for this problem. Include the appropriate distribution, critical value, and state where the null hypothesis can and cannot be rejected. f. Calculate the test statistic. g. What is your conclusion? Explain in detail. h. Find the p-value. Restate the conclusion.

Answer 1

a. H0: Proportion remains 41%. H1: Proportion increased. Use normal distribution, α = 0.05.

b. Normal distribution for large sample size (n = 100) is suitable.

c. Central limit theorem validates normal distribution approximation.

d. Significance level: 0.05.

e. Decision rule: Reject H0 if z > 1.645, otherwise, do not reject.

f. Test statistic: z = 2.84.

g. Conclusion: Reject H0; evidence suggests an increase.

h. P-value < 0.05, rejecting H0, indicating a likely increase.

Step-by-step explanation:

In this statistical hypothesis test, we assess whether there has been a change in the proportion of Americans desiring Internet access on their cell phones. The null hypothesis assumes that the proportion remains at 41%, while the alternative hypothesis posits an increase in this proportion. With a sample size of 100, employing a significance level of 0.05, the problem warrants the use of a normal distribution due to the sufficiently large sample size.

The central limit theorem validates this choice since the sample size meets the condition for approximating the sample proportion's distribution as normal. Calculating the test statistic using the sample proportion and hypothesized proportion, we find a z-value of 2.84. Comparing this to the critical value of 1.645 leads to the rejection of the null hypothesis.

The evidence supports the conclusion that the proportion of Americans seeking Internet access on their cell phones has likely increased. The p-value being less than 0.05 reinforces this conclusion, indicating a strong rejection of the null hypothesis. Therefore, there's substantial statistical evidence suggesting an increase in the desire for Internet access on cell phones among Americans.