Question

A report about how American college students manage their finances includes data from a survey of college students. Each person in a representative sample of 793 college students was asked if they had one or more credit cards and if so, whether they paid their balance in full each month. There were 500 who paid in full each month. For this sample of 500 students, the sample mean credit card balance was reported to be $825. The sample standard deviation of the credit card balances for these 500 students was not reported, but for purposes of this exercise, suppose that it was $200. Is there convincing evidence that college students who pay their credit card balance in full each month have a mean balance that is lower than $905, the value reported for all college students with credit cards

Answer 1

Answer:

Yes. There is enough evidence to support the claim that college students who pay their credit card balance in full each month have a mean balance that is lower than $905.

Step-by-step explanation:

We want to test the claim that college students who pay their credit card balance in full each month have a mean balance that is lower than $905.

To perform this test we have a sample of 500 students which have paid their balance in full each month. The sample mean is $825 and the estimated sample deviation is considered $200.

Then, the null and alternative hypothesis are:

$H_0: \mu=905\\\\H_a:\mu< 905$

The significance level is 0.05.

The sample has a size n=500.

The sample mean is M=825.

The estimated standard error of the mean is computed using the formula:

$s_M=\dfrac{s}{\sqrt{n}}=\dfrac{200}{\sqrt{500}}=8.94$

Then, we can calculate the t-statistic as:

$t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{825-905}{8.94}=\dfrac{-80}{8.94}=-8.94$

The degrees of freedom for this sample size are:

$df=n-1=500-1=499$

This test is a left-tailed test, with 499 degrees of freedom and t=-8.94, so the P-value for this test is calculated as (using a t-table):

$\text{P-value}=P(t<-8.94)=0$

As the P-value (0) 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 college students who pay their credit card balance in full each month have a mean balance that is lower than $905.