Question

In a simple random sample of 14001400 young​ people, 9090​% had earned a high school diploma. Complete parts a through d below. a. What is the standard error for this estimate of the percentage of all young people who earned a high school​ diploma? nothing ​(Round to four decimal places as​ needed.) b. Find the margin of​ error, using a​ 95% confidence​ level, for estimating the percentage of all young people who earned a high school diploma. nothing​% ​(Round to one decimal place as​ needed.) c. Report the​ 95% confidence interval for the percentage of all young people who earned a high school diploma. (nothing % comma nothing % )%,% ​(Round to one decimal place as​ needed.) d. Suppose that in the​ past, 80% of all young people earned high school diplomas. Does the confidence interval you found in part c support or refute the claim that the percentage of young people who earn high school diplomas has​ increased? Explain.

Answer 1

Answer:

(a) The standard error is 0.0080.

(b) The margin of error is 1.6%.

(c) The 95% confidence interval for the percentage of all young people who earned a high school diploma is (88.4%, 91.6%).

(d) The percentage of young people who earn high school diplomas has ​increased.

Step-by-step explanation:

Let p = proportion of young people who had earned a high school diploma.

A sample of n = 1400 young people are selected.

The sample proportion of young people who had earned a high school diploma is:

$\hat p=0.90$

(a)

The standard error for the estimate of the percentage of all young people who earned a high school​ diploma is given by:

$SE_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}}$

Compute the standard error value as follows:

$SE_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}}$

       $=\sqrt{\frac{0.90(1-0.90)}{1400}}$

       $=0.008$

Thus, the standard error for the estimate of the percentage of all young people who earned a high school​ diploma is 0.0080.

(b)

The margin of error for (1 - α)% confidence interval for population proportion is:

$MOE=z_{\alpha/2}\times SE_{\hat p}$

Compute the critical value of z for 95% confidence level as follows:

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

Compute the margin of error as follows:

$MOE=z_{\alpha/2}\times SE_{\hat p}$

          $=1.96\times 0.0080\\=0.01568\\\approx1.6\%$

Thus, the margin of error is 1.6%.

(c)

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

$CI=\hat p\pm MOE\\=0.90\pm 0.016\\=(0.884, 0.916)\\\approx (88.4\%,\ 91.6\%)$

Thus, the 95% confidence interval for the percentage of all young people who earned a high school diploma is (88.4%, 91.6%).

(d)

To test whether the percentage of young people who earn high school diplomas has​ increased, the hypothesis is defined as:

H₀: The percentage of young people who earn high school diplomas has not​ increased, i.e. p = 0.80.

Hₐ: The percentage of young people who earn high school diplomas has not​ increased, i.e. p > 0.80.

Decision rule:

If the 95% confidence interval for proportions consists the null value, i.e. 0.80, then the null hypothesis will not be rejected and vice-versa.

The 95% confidence interval for the percentage of all young people who earned a high school diploma is (88.4%, 91.6%).

The confidence interval does not consist the null value of p, i.e. 0.80.

Thus, the null hypothesis is rejected.

Hence, it can be concluded that the percentage of young people who earn high school diplomas has ​increased.