Question

1. the paralyzed veterans of america is a philanthropic organization that relies on contributions. they send free mailing labels and greeting cards to potential donors on their list and ask for voluntary contribution. to test a new campaign they recently sent letters to a random sample of 100,000 potential donors and received 4781 donations.

a) give a 95% 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 5%. given the confidence interval you found, do you find that percentage plausible?

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{4781}{100000}=0.04781$.

Step2: Calculate standard error

The standard error $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.04781\times(1 - 0.04781)}{100000}}\approx\sqrt{\frac{0.04781\times0.95219}{100000}}\approx\sqrt{\frac{0.04553}{100000}}\approx0.000675$.

Step3: Find critical value

For a 95% confidence interval, the critical value $z = 1.96$.

Step4: Calculate confidence interval

The lower bound is $\hat{p}-z\times SE=0.04781 - 1.96\times0.000675=0.04781-0.001323 = 0.046487$.
The upper bound is $\hat{p}+z\times SE=0.04781 + 1.96\times0.000675=0.04781 + 0.001323=0.049133$.

Answer:

a) The 95% confidence interval is $(0.0465, 0.0491)$.
b) Since $0.05$ is not within the confidence interval $(0.0465, 0.0491)$, the percentage of 5% is not plausible.