Question

Research suggests that about 60% of CSU graduates started at a community college. In random samples of 100 CSU graduates, the percentage who started at a community college varies. What is the probability that in a random sample of 100 CSU graduates the error is within 5% of the population proportion of 60%

Answer 1

Answer:

The probability that in a random sample of 100 CSU graduates the error is within 5% of the population proportion of 60% is 0.6923.

Step-by-step explanation:

According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.

The mean of this sampling distribution of sample proportion is:

 $\mu_{\hat p}=p$

The standard deviation of this sampling distribution of sample proportion is:

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

The information provided is:

p = 0.60

n = 100

As n = 100 > 30, the central limit theorem can be applied to approximate the sampling distribution of sample proportions.

The distribution of sample proportion is $\hat p\sim N(0.60, 0.049^{2})$.

Compute the probability that in a random sample of 100 CSU graduates the error is within 5% of the population proportion of 60% as follows:

$P(|\hat p-\mu_{\hat p}|=0.05)=P(-0.05$

                             $=P(\frac{-0.05}{0.049}$

Thus, the probability that in a random sample of 100 CSU graduates the error is within 5% of the population proportion of 60% is 0.6923.