Question

Research at the University of Toledo indicates that 50 percent ofthe students change their major area of study after their firstyear in a program. A random sample of 100 students in the Collegeof Business revealed that 48 had changed their major area of studyafter their first year of the program.Has there been a significant decrease in the proportion of studentswho change their major after the first year in this program? Testat the .05 level of significance.(a) What is the decisionrule? (Round youranswer to 3 decimal places.)(b) The value ofthe test statistic is . (Roundyour answer to 2 decimal places.)(c) What is yourdecision regarding ?

Answer 1

a) The decision rule is:

Reject the null hypothesis if P-value is under 0.05 (or the test statistic is larger than z=-1.645).

b) Test statistic z=-0.30

c) The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that there has been a significant decrease in the proportion of students who change their major after the first year in this program.

Explanation:

This is a hypothesis test for a proportion.

The claim we have to test is that there has been a significant decrease in the proportion of students who change their major after the first year in this program.

Then, the null and alternative hypothesis are:

`H_0: \pi=0.5\\\\H_a:\pi< 0.5`

The significance level is α=0.05.

The sample has a size n=100.

The sample proportion is p=0.48.

`p=X/n=48/100=0.48`

The standard error of the proportion is:

`\sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt{(0.5*0.5)/(100)}\\\\\\ \sigma_p=√(0.0025)=0.05`

We can calculate the test statistic as:

`z=(p-\pi-0.5/n)/(\sigma_p)=(0.48-0.5+0.5/100)/(0.05)=(-0.015)/(0.05)=-0.3`

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

`P-value=P(z<-0.3)=0.3821`

The P-value (0.3821) is bigger than the significance level (0.05), then the effect is not significant.

The null hypothesis failed to be rejected.

If we use the critical value approach, the critical value of z for this test with 5% significance level is z=1.645.

There is not enough evidence to support the claim that there has been a significant decrease in the proportion of students who change their major after the first year in this program.