Question

A camp accrediting association has 2,4002, comma, 400 camps in their association. They claim that 45%45, percent of their camps offer specialized programs for individuals with disabilities. Suppose you contact a simple random sample of 200

200 of the camps and ask them whether they offer specialized programs for individuals with disabilities.

Assuming that the association's claim is correct, what is the approximate probability that between 40%

40, per and 50%50, perc of the camps reply "yes" to your question?

Answer 1

Answer:

The probability that the proportion of campers who reply yes is between 40% and 50% is 0.8444.

Step-by-step explanation:

According to the Central limit theorem, if a large sample (n > 30) is selected from an unknown population then the sampling distribution of sample proportion follows a Normal distribution.

The mean of this distribution is:

$\mu_{\hat p}= p=0.45$

The standard deviation of this distribution is:

$\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}=0.0352$

Compute the probability that the proportion of campers who reply yes is between 40% and 50% as follows:

$P(0.40<p<0.50)=P(\frac{0.40-0.45}{0.0352}<\frac{p-\mu_{\hat p}}{\sigma_{\hat p}}<\frac{0.50-0.45}{0.0352})\\=P(-1.42<Z<1.42)\\=P(Z<1.42)-P(Z<-1.42)\\=P(Z<1.42)-1+P(Z<1.42)\\=2P(Z<1.42)-1\\=(2\times0.9222)-1\\=0.8444$

Thus, the probability that the proportion of campers who reply yes is between 40% and 50% is 0.8444.