Answer to Question #219922 in Statistics and Probability for sai

Question

According to a​ newspaper, 59​% of high school seniors have a​ driver's license. Suppose we take a random sample of 100 high school seniors and find the proportion who have a​ driver's license.

a. What value should we expect for our sample​ proportion?

b. What is the standard​ error?

c. Use your answers to parts​ (a) and​ (b) to complete this​ sentence:

We expect​ ____% to have their​ driver's license, give or take​ _____%.

d. Suppose we increased the sample size from 100 to 300. What effect would this have on the standard​ error? Recalculate the standard error to see if your prediction was correct.

Answer 1

a.

$$E(\bar{p})=p=0.59$$

b.

$$se(\bar{p})=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.59(1-0.59)}{100}}\approx0.0492$$

c. We expect​ $59$ % to have their​ driver's license, give or take $4.92$ %.

d. If we increased the sample size from 100 to 300, then the standard error will decrease by a factor of $\sqrt{3}$ .

$$E(\bar{p})=p=0.59$$

$$se(\bar{p})=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.59(1-0.59)}{300}}\approx0.0284$$

We expect​ $59$ % to have their​ driver's license, give or take $2.84$ %.