Question

Asked Jul 29, 2019

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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 68 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.)

Give a brief interpretation of the meaning of the confidence interval you have found.

(c) Do you think the conditions*np* > 5 and *nq* > 5 are satisfied in this problem? Explain why this would be an important consideration.

(b) Find a 95% confidence interval for

lower limit | |

upper limit |

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 95% confident that the true proportion of actors who are extroverts falls within this interval.We are 5% confident that the true proportion of actors who are extroverts falls above this interval. We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.We are 5% confident that the true proportion of actors who are extroverts falls within this interval.

(c) Do you think the conditions

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial. Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately

Step 1

**(a) Point estimation for p:**

It is given that out of 68 professional actors, 40 are extroverts.

The point estimate (*p*-hat) is calculated as **0.5882**, from the calculations given below.

Step 2

**(b) Confidence interval for p:**

Here, the sample size, *n* = 68.

The probability of success *p* = 0.40

The probability of failure *q* = 0.60 (= 1 – 0.40)

The value of *np* = 27.2 (=68×0.40)

The value of *nq* = 40.8 (=68×0.60)

Since *np* and *nq* are greater than 5, *p*-hat is approximately normal.

The formula for confidence interval for the population proportion when *np* and *nq* are greater than 5 is as follows.

Step 3

**Computation of the 95% confidence interval for p:**

Here, the sample proportion (*p*-hat) is 0.5882.

The confidence level is 0.95.

Thus, the level of significance, *α* is 0.05 (= 1 – 0.95).

For 95% confidence interval, the critical value for *z*0.025 ...

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