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 38 were extroverts. A button hyperlink to the SALT program that reads: Use SALT. (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.) Incorrect: Your answer is incorrect. (b) Find a 95% confidence interval for p. (Round your answers to two decimal places.) lower limit Incorrect: Your answer is incorrect. upper limit Incorrect: Your answer is incorrect.

Answer 1

a) 0.6032

b)

Lower limit: 0.48

Upper limit: 0.72

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

Question a:

In a random sample of 63 professional actors, it was found that 38 were extroverts.

We use this to find the sample proportion, which is the point estimate for p. So

`\pi = (38)/(63) = 0.6032`

Question b:

Sample of 63 means that
`n = 63`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.6032 - 1.96\sqrt{(0.6032*0.3968)/(63)} = 0.4824`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.6032 + 1.96\sqrt{(0.6032*0.3968)/(63)} = 0.724`

Rounding to two decimal places:

Lower limit: 0.48

Upper limit: 0.72