Question

(G Twenty years ago, a very tamous Psychologist Specializing in manage counseling autized bouk detailing the nay in which she believed spouses Should communicate. She is now interested in the proportion of all couples who bought her book who stayed together. For a random sample of 300- couples who bought her book, she found mat 228 of them stayed together. Based on this, compute a 95% confidence interval for the proportion of all couples who bought her book who stayed together, then compute me table below. Carry your intermediate computations to at least three Round your answers to pigus. two decimal decimal places What is the lower limit of me 951. confidence interval? 951. confidence what is the upper limit of the internal ? F

Answer 1

Solution :

Given that,

n = 300

x = 228

Point estimate = sample proportion = $\hat p$ = x / n = 228 / 300 = 0.760

1 - $\hat p$ = 1 - 0.760 = 0.240

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z_{0.025 = 1.960}

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$ (($\hat p$ * (1 - $\hat p$ )) / n)

= 1.96 ($\sqrt$((0.760 * 0.240)) / 300)

= 0.048

A 95% confidence interval for population proportion p is ,

$\hat p$ - E < p < $\hat p$ + E

0.760 - 0.048 < p < 0.760 + 0.048

0.712 < p < 0.808

( 0.712 , 0.808 )

The Lower limit of the 95% confidence interval is : 0.71

The Upper limit of the 95% confidence interval is : 0.81