Question

A philanthropic organisation sent free mailing labels and greeting cards to a random sample of 100 comma 000100,000 potential donors on their mailing list and received 47034703 donations. ​(a) Give a 9999​% confidence interval for the true proportion of those from their entire mailing list who may donate. ​(b) A staff member thinks that the true rate is 4.74.7​%. Given the confidence interval you​ found, do you find that rate​ plausible?

Answer 1

Answer:

a) 99% Confidence interval:  (0.045311,0.04875)

b) 4.7% is a plausible rate.

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 100,000

Number of people who donated, x = 4703

$\hat{p} = \dfrac{x}{n} = \dfrac{4703}{100000} = 0.04703$

a) 99% Confidence interval:

$\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

$z_{critical}\text{ at}~\alpha_{0.01} = 2.58$

Putting the values, we get:

$0.04703\pm 2.58(\sqrt{\dfrac{0.04703(1-0.04703)}{100000}})\\\\= 0.04703\pm 0.00172\\\\=(0.04531,0.04875)$

b) A staff member thinks that the true rate is 4.74.7​%.

Yes, this is a plausible rate as this value of 0.0474 lies in the calculated confidence interval as:

$4.531\%$