Question

The average cost of tuition plus room and board at small private liberal arts colleges is reported to be less than $18,500 per term. A financial administrator at one of the colleges believes that the average cost is higher. The administrator conducted a study using 150 small liberal arts colleges. It showed that the average cost per term is $18,200. The population standard deviation is known to be $1,400. Let α= 0.05. What are the null and alternative hypothesis for this study?

Answer 1

The null and alternative hypothesis for this study are:

`H_0: \mu=18500\\\\H_a:\mu< 18500`

The null hypothesis is rejected (P-value=0.004).

There is enough evidence to support the claim that the average cost of tuition plus room and board at small private liberal arts colleges is less than $18,500 per term.

Explanation:

This is a hypothesis test for the population mean.

The claim is that the average cost of tuition plus room and board at small private liberal arts colleges is less than $18,500 per term.

Then, the null and alternative hypothesis are:

`H_0: \mu=18500\\\\H_a:\mu< 18500`

The significance level is 0.05.

The sample has a size n=150.

The sample mean is M=18200.

The standard deviation of the population is known and has a value of σ=1400.

We can calculate the standard error as:

`\sigma_M=(\sigma)/(√(n))=(1400)/(√(150))=114.31`

Then, we can calculate the z-statistic as:

`z=(M-\mu)/(\sigma_M)=(18200-18500)/(114.31)=(-300)/(114.31)=-2.624`

This test is a left-tailed test, so the P-value for this test is calculated as:

`P-value=P(z<-2.624)=0.004`

As the P-value (0.004) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the average cost of tuition plus room and board at small private liberal arts colleges is less than $18,500 per term.