Question

Suppose that you decide to randomly sample people ages 18-24 in your county to determine whether or not they are registered to vote. In your sample of 50 people, 35 said they were registered to vote. a) (2 points) Find a 95% confidence interval for the true proportion of the county population ages 18-24 who are registered to vote. Make sure to check any necessary conditions and to state a conclusion in the context of the problem. Also, explain what 95% confidence means in this context. b) (1 point) What is the probability that the true proportion of people ages 18-24 who registered to vote in your county is in your particular confidence interval? (Note: Be careful). c) (1 point) According to a separate news report, about 73% of 18- to 24-year-olds in the same county said that they were registered to vote. Does the 73% figure seem reasonable with your own poll? Explain. d) (1 point) Assume you have not done your poll yet, but you knew the news report poll results. In designing your poll now, you want separately estimate the same percentage to within ±4 percentage points with 95% confidence, how many people should you poll?

Answer 1

a) sample proportion ,

for 95% CI , z_c = 1.96

95% confidence interval for true proportion

= 0.7 0.1270

= (0.5730, 0.8270)

conditions for 95% confidence interval using standard normal distribution

as np = 50 * 0.7 = 35 > 15

n is large and p is nearly close to 0.5

by 95% confidence interval we mean that , we 95% confident that the true proportion of registered voter in the age group 18-24 lies in the interval 0.5730 to 0.8270 or 57.3% to 82.7%

b) there is 0.95 probability that true proportion will lie in the interval ( 0.5730, 0.8270)

C) YES , it seems reasonable as 73% lies in between the 95% confidence interval

d) margin of error

solving n = 473.24

we should poll 473 people