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 67 professional actors, it was found that 40 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

Answer: a. A point estimate for p is 0.597 .

b. The 95% confidence interval for p.

Lower limit = 0.48

Upper limit = 0.71

Step-by-step explanation:

Given : Sample size of professional actors : n= 67

Number of extroverts : x= 40

Let p represent the proportion of all actors who are extroverts.

a. The point estimate for p = sample proportion = $\hat{p}=\dfrac{x}{n}$

$=\dfrac{40}{67}=0.597$

b. Confidence interval for population proportion :

$\hat{p}\pm z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

Since the critical value for 95% confidence interval is 1.96 , so the 95% confidence interval for p would be

$0.597\pm (1.96)\sqrt{\dfrac{0.597(1-0.597)}{67}}$

$0.597\pm (1.96)\sqrt{0.0036}$

$0.597\pm (1.96)(0.06)$

$0.597\pm 0.1176$

$(0.597-0.1176,\ 0.597+0.1176)\\\\=(0.4794,\ 0.7146)\\\\\approx(0.48,\ 0.71)$

In the 95% confidence interval for p.

Lower limit = 0.48

Upper limit = 0.71

Answer 2

Answer: the upper limit is 0.58

The lower limit is 0.56

Step-by-step explanation:

Total number of professional actors that was sampled is 67. it was found that 40 were extroverts.

a) the proportion of all actors who are extroverts would be

p = 40/70 = 0.5714

Therefore, the proportion of all actors who are not extroverts would be

q = 1 - p = 1 - 0.5714 = 0.4286

b) The formula for determining the confidence interval for a population proportion is expressed as

p ± z[p(1 - p)/n

The z value for a 95% confidence level is 1.96.

Therefore

Confidence interval

= 0.5714 ± 1.96[0.5714(1 - 0.5714)/67

= 0.5714 ± 1.96(0.2449/67)

= 0.5714 ± 0.00716

The upper limit would be

0.5714 + 0.00716 = 0.58

The lower limit would be

0.5714 - 0.00716 = 0.56