Question

You would like to estimate a population proportion p. For example, this might be the proportion of all undergraduate students who own iPhones. You take a sample of size n and discover the sample proportion p̂, the proportion of students in the sample who own iPhones. Assume that p=0.36 and n=500. Which of the following is true?

a. If you took many samples of size 500 from this population and calculated a sample proportion p̂ for each, the standard deviation of these p̂ 's would be about 0.021.

b. If you took many samples of size 500 from this population and calculated a p̂ for each, and you then created a histogram of the p̂ 's, it would be approximately normal with mean close to 0.36 and standard deviation equal close to 0.021.

c. If you took many samples of size 500 from this population and calculated a p̂ for each, it would not be very surprising to see some of the p̂ 's below 0.34 or above 0.38.

d. All of these choices are true.

Answer 1

Solution :

Given that,

n = 500

Point estimate = sample proportion = =0.36

1 - = 1-0.36 = 0.64

= p = 0.36

=  [p( 1 - p ) / n] = [0.36*(0.64) /500 ] = 0.021

Option a is correct answer

a. If you took many samples of size 500 from this population and calculated a sample proportion p̂ for each, the standard deviation of these p̂ 's would be about 0.021