Question

In a pool of n =1000 randomly selected teenagers, 450 indicated that they like Netflix,

a) Construct a 95% confidence interval for the proportion of teenagers who like Netflix and interpret it.

b) Why the result in a) is approximately valid.

c) If you wish to estimate the proportion of teenagers who like Netflix correct to within 0.025 with 95% confidence, how large should the sample size n be?

please write all the formulas in your solution. Thank you.

Answer 1

Solution:

Given:

Sample size = n = 1000

x = Number of  teenagers indicated that they like Netflix = 450

Part a) Construct a 95% confidence interval for the proportion of teenagers who like Netflix and interpret it.

Formula:

where

and

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Thus

Thus we are 95% confident that the true population proportion of teenagers who like Netflix is between 0.4192 and 0.4808.

Part b) Why the result in a) is approximately valid.

Since sample is collected randomly and sample size n = 1000 is large enough so that

and

thus sampling distribution of sample proportions is approximately Normal.

Thus the result in a) is approximately valid.

Part c)

E = Margin of Error = 0.025

c = confidence level = 95%

p = proportion estimate =

Formula for sample size n:

.