A bank wonders whether omitting the annual credit card fee for customers who charge ar least $3000 in a year will increase the amount charged on its cards. The bank makes this offer to an SRS of 400 of its customers. It then compares how much these customers charge this year with the amount they charged last year. The mean increase in the sample $246 and the standard deviation is $112. Is there significant evidence at the 1% level that the mean amount charged increases under the no-fee offer? State Ho and Ha and carry out a significance test. Use significance level 0.01

Question

A bank wonders whether omitting the annual credit card fee for customers who charge ar least $3000 in a year will increase the amount charged on its cards. The bank makes this offer to an SRS of 400 of its customers. It then compares how much these customers charge this year with the amount they charged last year. The mean increase in the sample $246 and the standard deviation is $112. Is there significant evidence at the 1% level that the mean amount charged increases under the no-fee offer? State Ho and Ha and carry out a significance test. Use significance level 0.01

Answer 1

To determine whether there is significant evidence at the 1% level that the mean amount charged increases under the no-fee offer, we need to set up the null hypothesis (Ho) and the alternative hypothesis (Ha).

Ho: The mean amount charged with the no-fee offer (μ) is equal to or less than the mean amount charged without the no-fee offer (μ0).
Ha: The mean amount charged with the no-fee offer (μ) is greater than the mean amount charged without the no-fee offer (μ0).

To carry out a significance test, we can use a one-sample t-test. The t-test compares the mean difference to a hypothesized value and calculates a t-statistic. We then compare the t-statistic to the critical value from the t-table to determine statistical significance.

Here are the steps to perform the significance test:

Step 1: Set up the null (Ho) and alternative (Ha) hypotheses.
Ho: μ <= μ0
Ha: μ > μ0

Step 2: Determine the significance level (alpha).
The significance level (α) is given as 0.01 or 1%.

Step 3: Calculate the test statistic.
The test statistic for a one-sample t-test is calculated using the following formula:

t = (x̄ - μ0) / (s / sqrt(n))

where:
x̄ is the sample mean increase in charges ($246),
μ0 is the hypothesized mean increase in charges under the no-fee offer,
s is the standard deviation of the sample ($112), and
n is the sample size (400).

Substituting the values into the formula, we get:
t = (246 - μ0) / (112 / sqrt(400))

Step 4: Determine the critical value.
Looking up the critical value corresponding to a one-sided test with a significance level of 0.01 and a degrees of freedom (df) = n - 1 = 399, we find that the critical value is approximately 2.617.

Step 5: Compare the test statistic with the critical value.
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 6: Make a conclusion.
If we reject the null hypothesis, there is significant evidence at the 1% level that the mean amount charged increases under the no-fee offer. If we fail to reject the null hypothesis, we do not have enough evidence to conclude that there is a significant increase in mean charges.

Since the test statistic (t) is not given in the question, you need to calculate it using the formula in Step 3 and then compare it with the critical value in Step 4 to reach a conclusion.