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 514 judges, it was found that 276 were introverts.

(a) Let p represent the proportion of all judges who are introverts. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answers to two decimal places.)

lower limit
upper limit

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

We are 1% confident that the true proportion of judges who are introverts falls within this interval.

We are 1% confident that the true proportion of judges who are introverts falls above this interval.    

We are 99% confident that the true proportion of judges who are introverts falls within this interval.

We are 99% confident that the true proportion of judges who are introverts falls outside this interval.

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

Yes, the conditions are 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 normal.    

No, the conditions are not 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 normal.

Answer 1

Solution :

Given that,

n = 514

x = 276

a) Point estimate = sample proportion = = x / n = 276 / 514 = 0.5370

1 - = 1 - 0.5370 = 0.4630

b) At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

Z/2 = Z_{0.005 = 2.576}

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

= 2.576 (((0.5370 * 0.4630) / 514)

= 0.06

A 99% confidence interval for population proportion p is ,

± E

= 0.5370 ± 0.06

= ( 0.48, 0.60 )

lower limit = 0.48

upper limit = 0.60

We are 99% confident that the true proportion of judges who are introverts falls within this interval.

c) Yes, the conditions are satisfied, This is important because it allows us to say that is approximately normal.