Question

A bank wonders if omitting the annual credit card fee for customers who charge at least $\backslash (2400$in a year will increase the amount charged on its credit cards. The bank makes this offer to an SRS of $200$of its credit card customers. It then compares how much these customers charge this year with the amount that they charged last year. The mean increase in the sample is $\backslash )332$, and the standard deviation is$\$108$.

a. Construct and interpret a $99\%$confidence interval for the true mean increase.

b. Based on the interval from (a), can you conclude that dropping the annual fee would cause an increase in the average amount spent by this bank’s credit card customers? Why or why not?

Short answer

Part (a) We are $95\%$confident that the true mean increases is between $311.8956and352.1044$.

Part (b) No, we cannot conclude.

Step-by-step solution

Part (a) Step 1. Given information

It is given:

$$\begin{gathered} \overline{x}=332 \\ n=200 \\ s=108 \\ c=0.99 \end{gathered}$$

Part (b) Step 2. Calculation

The degree of freedom will be:

$$\begin{gathered} df=n-1 \\ =200-1 \\ =199 \end{gathered}$$

Now, the value of t will be:

$t_{\alpha /2}=2.626$

Now, the confidence interval will be:

$$\begin{gathered} \overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}} \\ =332-2.626\times \frac{108}{\sqrt{200}} \\ =311.8956 \\ =\overline{x}+t_{\alpha /2}\times \frac{s}{\sqrt{n}} \\ =332+2.626\times \frac{108}{\sqrt{200}} \\ =352.1044 \end{gathered}$$

Thus we conclude that we are $99\%$ confident that the true mean increases is between$311.8956and352.1044$.

Part (b) Step 1. Explanation

A completely randomized experiment randomly assigns all subjects to a group.

In this case, the experiment is not completely randomized since we use the subjects before and after the treatment. Thus the experiment is not a completely randomized experiment.

We cannot prove causation if we do not use a completely randomized experiment because it is possible that the causation of the difference is another variable.

For example, inflation could be the cause because everything you buy will be slightly more expensive the next year due to inflation and thus the increase in the amount charged could be due to inflation.