Question

According to the Institute for Students in Shackles, 70% of all college students in a recent year graduated with student loan debt.  The University of Florida reports that only 52% of its graduates from a random sample of 500 students have student loan debt. Use a hypothesis test to determine if there is enough evidence to support UF’s claim that student loan debt is less.

a) State your null and alternative hypothesis.

b) Find p-hat, SD, Z, and the P-value

Answer 1

Let p be the true proportion of University of Florida students who graduated with student loan debt. We want to test the claim of UF that student loan debt of UF students is less than the overall figure of 0.70.

That is we want to test if p<0.70

a) The hypotheses are

b) We have the following sample information

n=500 is the sample size of the UF students sampled

is the sample proportion of UF students graduated with loan debt

ans: p-hat = 0.52

is the hypothesized value of the proportion of UF students graduated with loan debt (from the null hypothesis)

is the standard error of proportion

ans: SD = 0.0205

The value of are both greater than 5. Hence we can use the normal approximation for the distribution of proportion

The test statistics is

ans: Z=-8.78

This is a one tailed (left tailed) test (The alternative hypothesis has "<")

The p-value is the are under the curve of the left tail.

p-value=P(Z<-8.78).

This is same as P(Z>8.78) or 1-P(Z<8.78)

Using the standard normal table we can get P(Z<8.78) = 1.0000 or P(Z<-8.78) = 1-1.000=0.0000

ans: p-value=0.0000

The exact p-value can be got in excel using =NORM.DIST(-8.78,0,1,TRUE). p-value=8.17e-19

Finally we will reject the null hypothesis is the p-value is less than the significance level of the test.

Here the p-value of 0.0000 is less than the significance level of 0.05

Hence we reject the null hypothesis.

We conclude that there is sufficient evidence to support UF’s claim that student loan debt is significantly less (than 0.70)