Question

What is the proportion of college students that use the college recreational facilities more than three times per week? Out of a sample of 464 college students, 116 indicated that they used the college recreational facilities more than three times per week. Calculate a confidence interval at the 99% confidence level following the steps below.

a)Write down the symbol used to represent the sample’s proportion as well as its value.

b) Calculate the margin of error for the 99% confidence interval. Show all your work and round your answer to 3 decimal places (need both correct equation, and correct answer for full credit).

c)Write down the lower and upper bounds of the 99% confidence interval. Write your answer in the blanks below.

( ______ , ______)

Answer 1

Sample proportion = = x / n

Where x = Number of college students that use the college recreational facilities more than three times per week.

n = sample size.

   = 116 / 464 = 0.25

b ) Formula of margin of error is as follows:

..........( 1 )

It is given that ; c = confidence level = 0.99

so that level of significance = = 1 - c = 1 - 0.99 = 0.01

this implies that \alpha/2 = 0.01/2 = 0.005

So we want to find such that  

P(Z > ) = 0.005

Therefore ,

P(Z < ) = 1 - 0.005 = 0.995

The general excel command to find critical z value is

"=NORMSDIST(probability)"

Here probability = 0.995

So that critical is = "=NORMSINV(0.995)" = 2.5758

also n = 464 and = 0.25

Put this value in equation (1) so we get

So E = 0.052

Lower limit = - E = 0.25 - 0.052 = 0.198

Upper Limit =    + E = 0.25 + 0.052 = 0.302

So that 99% confidence interval for population proportion P is (0.198 , 0.302)