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

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.

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{(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<\hat p-\mu_(\hat p)<0.05)`

`=P((-0.05)/(0.049)<(\hat p-\mu_(\hat p))/(\sigma_(\hat p))<(0.05)/(0.049))\\\\=P(-1.02<Z<1.02)\\\\=P(Z<1.02)-P(Z<-1.02)\\\\=0.84614-0.15386\\\\=0.69228\\\\\approx 0.6923`

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.