Question 1 Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your metho

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Question 1 Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your metho

ds as well as on the accuracy and completeness of your results and explanations. A recent survey collected information on television viewing habits from a random sample of 1,000 people in the United States. Of those sampled, 37 percent indicated that their favorite sport to watch on television was American football.
(a) Construct and interpret a 95 percent confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football.
(b) Based on your answer to part (a), is it reasonable to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football? Justify your answer.

Mathematics

1 answer:

babunello [11.8K] · Answer 1 · 2026-02-18T09:23:22+03:00

Answer:

(a) The 95% confidence interval representing the percentage of the entire U.S. population that would select American football as their preferred television sport is (0.34, 0.40).

(b) Not reasonable.

Step-by-step explanation:

Given:

n = 1000

$\hat p$ = 0.37

(a)

The confidence interval (1 - α)% for proportion p is:

$CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}$

Where:

$\hat p$ = the sample proportion

n = sample size

$z_{\alpha/2}$ = critical z value.

Calculate the critical value of z for a 95% confidence level as shown:

$z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96$

*Refer to a z-table for the necessary value.

Compute the 95% confidence interval for proportion p as follows:

$CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}$

$=0.37\pm 1.96\times\sqrt{\frac{0.37(1-0.37)}{1000}}$

$=0.37\pm 0.03\\=(0.34, 0.40)$

Thus, the 95% confidence interval indicating the proportion of individuals in the U.S. who might say their preferred sport on TV is American football is (0.34, 0.40).

(b)

Next, we must assess if it's rational to consider that the actual percent of people in the United States who favor American football on television is 33%.

The hypothesis is defined as:

H₀: The portion of the U.S. population claiming American football as their favorite sport on television is 33%, meaning p = 0.33.

Hₐ: The proportion of people in the U.S. whose favorite sport to watch on television is not 33%, or p ≠ 0.33

This hypothesis can be verified using a confidence interval.

The decision rule:

If the (1 - α)% confidence interval contains the null value of the hypothesis, then the null hypothesis is not rejected. If, however, the (1 - α)% confidence interval excludes the null value of the hypothesis, then the null hypothesis is rejected.

The confidence interval does encompass the null value of p, which is 0.33.

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