Question

Directions: Show ALL work! Please round any answer to the nearest hundredth, unless you can get an exact amount. If an answer asks to explain, please make sure to give a valid and complete explanation. For your hypothesis tests, make sure you give a complete interpretation.

2. In a survey of 500 randomly selected U.S. teens, it was found that 37% of them have been in the car while a person under the influence is driving.

a) Construct a 90% confidence interval for the population proportion of U.S. teens who have been in the car while a person under the influence is driving.

b) Using the information from part a, determine the minimum sample size required to be 95% confident that the estimate is accurate within 3% of the population proportion.

Answer 1

Solution :

Given that,

a) n = 500

= 37% = 0.37

1 - = 1 - 0.37 = 0.63

At 90% confidence level

= 1 - 90%

=1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z_0.05 = 1.645

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 (((0.37 * 0.63) / 500 )

= 0.036

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.37 - 0.036 < p < 0.37 + 0.036

(0.334 < p < 0.406)

b) margin of error = E = 3% = 0.03

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = 1.96

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (1.96 / 0.03)² * 0.37 * 0.63

= 994.97

sample size = n = 995