A student at a four-year college claims that mean enrollment at two-year colleges is lower than at four-year colleges in the United States. Two surveys are conducted. Of the 35 two-year colleges surveyed, the mean enrollment was 5,068 with a standard deviation of 4,777. Of the 35 four-year colleges surveyed, the mean enrollment was 5,466 with a standard deviation of 8,191. Conduct a hypothesis test at the 5% significance level.

The p-value of the test is of 0.0708 > 0.05, which means that there is not enough evidence for the claim the the mean enrollment at two-year colleges is lower than at four-year colleges in the United States.

Step-by-step explanation:

To solve this question, we need to understand subtraction of normal variables and the central limit theorem.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation

Subtraction between normal variables:

When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

35 two-year colleges surveyed, the mean enrollment was 5,068 with a standard deviation of 4,777.

This means that

Of the 35 four-year colleges surveyed, the mean enrollment was 5,466 with a standard deviation of 8,191.

This means that

A student at a four-year college claims that mean enrollment at two-year colleges is lower than at four-year colleges in the United States.

At the null hypothesis, we test if is at least equal, that is, the subtraction of the mean enrollment at 2 years colleges subtracted by the mean enrollment at 4 years colleges is at least 0. So

At the alternative hypothesis, we test if is less, that is, the subtraction is less than 0.

The test statistic is:

In which X is the sample mean, is the value tested at the null hypothesis and s is the standard error.

0 is tested at the null hypothesis:

This means that

From the two samples:

Test statistic:

P-value of the test:

The p-value of the test is the probability of a difference of 398 or more, which is the p-value of z = -1.47.

Looking at the z-table, z = -1.47 has a p-value of 0.0708.

The p-value of the test is of 0.0708 > 0.05, which means that there is not enough evidence for the claim the the mean enrollment at two-year colleges is lower than at four-year colleges in the United States.

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