Question

The average undergraduate cost for tuition, fees, and room and board for two-year institutions last year was $13,252. The following year, a random sample of 20 two-year institutions had a mean of $15,560 and a standard deviation of $3500. Is there sufficient evidence at the ?= 0.05 level to conclude that the mean cost has increased. Solve the question by traditional approach.

Answer 1

The p-value of the test is 0.0041 < 0.05, which means that there is sufficient evidence at the 0.05 significance level to conclude that the mean cost has increased.

Explanation:

The average undergraduate cost for tuition, fees, and room and board for two-year institutions last year was $13,252. Test if the mean cost has increased.

At the null hypothesis, we test if the mean cost is still the same, that is:

`H_0: \mu = 13252`

At the alternative hypothesis, we test if the mean cost has increased, that is:

`H_1: \mu > 13252`

The test statistic is:

We have the standard deviation for the sample, which means that the t-distribution is used to solve this question

`t = (X - \mu)/((s)/(√(n)))`

In which X is the sample mean,
`\mu` is the value tested at the null hypothesis, s is the standard deviation and n is the size of the sample.

13252 is tested at the null hypothesis:

This means that
`\mu = 13252`

The following year, a random sample of 20 two-year institutions had a mean of $15,560 and a standard deviation of $3500.

This means that
`n = 20, X = 15560, s = 3500`

Value of the test statistic:

`t = (X - \mu)/((s)/(√(n)))`

`t = (15560 - 13252)/((3500)/(√(20)))`

`t = 2.95`

P-value of the test and decision:

The p-value of the test is found using a t-score calculator, with a right-tailed test, with 20-1 = 19 degrees of freedom and t = 2.95. Thus, the p-value of the test is 0.0041.

The p-value of the test is 0.0041 < 0.05, which means that there is sufficient evidence at the 0.05 significance level to conclude that the mean cost has increased.