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 63 professional actors, it was found that 41 were extroverts.

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

p̂=

q̂=

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

Find the maximal margin of error. (Round your answer to two decimal places.)

E =

Report the bounds from the 95% confidence interval for p. (Round your answers to two decimal places.)

lower limit =

upper limit =

Answer 1

Solution :

Given that,

n = 63

x = 41

Point estimate = sample proportion = = x / n = 41 / 63=0.6508

q =1 - = 1 - 0.6508=0.3492

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.96}

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

= 1.96 (((0.6508*0.3492) / 63)

= 0.12

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.6508 -0.12 < p <  0.6508 +0.12

0.53< p < 0.77

The 95% confidence interval for the population proportion p is :0.53 , 0.77