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: the upper limit is 0.58

The lower limit is 0.56

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

Answer 2

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

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}=(x)/(n)`

`=(40)/(67)=0.597`

b. Confidence interval for population proportion :

`\hat{p}\pm z^*\sqrt{\frac{\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{(0.597(1-0.597))/(67)}`

`0.597\pm (1.96)√(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