Question

The average undergraduate cost for tuition, fees, room, and board for all institutions last year was $26,025. A random sample of 40 institutions of higher learning this year indicated that the mean tuition, fees, room, and board for the sample was $27,690, and the population standard deviation is $5492. At the 0.05 level of significance, is there sufficient evidence that the cost has increased? Use P-Value Method. Find the 95% confidence level of the true mean. Does the confidence interval interpretation agree with the results of the hypothesis test?

Answer 1

the null hypothesis:

$\mu=26025$ , the mean cost is the same

the alternative hypothesis:

$\mu >26025$ , the mean cost has increased

$t=\frac{\overline{x}-\mu}{\sigma/\sqrt n}=\frac{27690-26025}{5492/\sqrt{40}}=1.917$

$df=40-1=39$

using a t-score calculator (www.socscistatistics.com):

p-value = 0.0313

Since p-value < 0.05, we reject the null hypothesis. So, the cost has increased.

95% confidence level of the true mean:

$-t_{0.05}<\frac{\mu-26025}{5492/\sqrt{40}}<t_{0.05}$

$-2.02<\frac{\mu-26025}{5492/\sqrt{40}}<2.02$

$24271<\mu <27779$

The confidence interval interpretation does not agree with the results of the hypothesis test, because the sample mean $27,690 is inside the confidence interval. This means that the mean cost is the same, not increased.