Question

A credit card company lowered its annual interest rate recently. Records showed that before the rate change, the average outstanding balance on a credit card was $580. The managers believed that if they lowered interest rates that this would increase the average outstanding balance. To test this claim, a random sample of 30 accounts was examined after the rate decrease. The average outstanding balance of this sample was $620 with a standard deviation of $105. At a 5% significance level, can it be concluded that the lower interest rate resulted in average account balance greater than $580?

a) Set up the null and alternative hypotheses in words and in mathematical symbols.

b) State the sample size, sample mean and sample standard deviation.

c) State the mean and standard error for the sampling distribution. Sketch the sampling distribution and label the mean, standard error, the observed value, and shade in the region for the P-value.

d) Find the test statistic, P-value, and state your conclusion in complete sentence(s).

Answer 1

a) As we are testing here whether the mean is greater than 580, therefore the null and the alternative hypothesis here are given as:

b) The sample measures here are given as:

is the sample size

is the sample mean

is the sample standard deviation here.

c) The mean and standard error of sampling distribution of sample mean here are computed as:

This is plotted on graph as:

The shaded region in the above graph is the p-value here that is the probability of having the sample mean more than 620.

d) The test statistic here is computed as:

Therefore 2.0866 is the required test statistic value here.

For n - 1 = 29 degrees of freedom, the p-value is computed from the t distribution tables here as:

p = P( t29 > 2.0866) = 0.0229

Therefore 0.0229 is the required p-value here.

As the p-value here is 0.0229 < 0.05 which is the level of significance here, therefore the test is significant here and we can reject the null hypothesis here and conclude that we have sufficient evidence that the lower interest rate resulted in average account balance greater than $580