Question

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 69 professional actors, it was found that 42 were extroverts.

(a) Let p represent the proportion of all actors who are extroverts. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 95% confidence interval for p. (Round your answers to two decimal places.)

lower limit
upper limit

Answer 1

Solution :

Given that,

n = 69

x = 42

Point estimate = sample proportion = $\hat p$ = x / n = 42 / 69=0.6087

1 - $\hat p$ = 1- 0.6087=0.3913

At 95% confidence level

$\alpha$ = 1 - 95%  

$\alpha$ = 1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z_{0.025 = 1.96}

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

= 1.96 ($\sqrt$((0.6087*0.3913) / 69)

= 0.1152

A 95% confidence interval for population proportion p is ,

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

0.6087 -0.1152 < p < 0.6087 + 0.1152

0.4935< p < 0.7239

The 95% confidence interval for the population proportion p is : (lower limit =0.49 , upper limit =0.72)