To solve this question, we will go through each of the provided steps in detail:
### a. State the hypotheses
The null hypothesis (H0) and the alternative hypothesis (H1) are:
- [tex]\( H_0: \mu_1 = \mu_2 \)[/tex] (The average credit card debt has not increased)
- [tex]\( H_1: \mu_1 < \mu_2 \)[/tex] (The average credit card debt has increased)
### b. Find the critical value
We are performing a one-tailed test with a significance level [tex]\( \alpha = 0.05 \)[/tex]. To find the critical value at [tex]\( \alpha = 0.05 \)[/tex], we look up the value from the standard normal distribution table (z-table):
[tex]\[ z_{0.95} = 1.6448536269514722 \][/tex]
### c. Compute the test statistic
Here we calculate the test statistic [tex]\( z \)[/tex] using the following formula:
[tex]\[ z = \frac{\text{mean difference}}{\text{standard error}} \][/tex]
Where:
- Mean difference = [tex]\( 8624 - 8021 = 603 \)[/tex]
- Standard error = [tex]\( \frac{\sigma}{\sqrt{n}} = \frac{1083}{\sqrt{2 \times 32}} = 271.5 \)[/tex]
So,
[tex]\[ z = \frac{603}{271.5} = 2.227146814404432 \][/tex]
### d. Make the decision
We compare the test statistic [tex]\( z = 2.23 \)[/tex] with the critical value [tex]\( 1.65 \)[/tex]:
Since [tex]\( 2.23 > 1.65 \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].
### e. Summarize the results
Based on our analysis, there is enough evidence to conclude that the average credit card debt has increased.
### Conclusion
Given the computations and comparison with the possible answers:
- The correct answer is B:
- a. [tex]\( H_0: \mu_1 = \mu_2 \)[/tex]; [tex]\( H_1: \mu_1 < \mu_2 \)[/tex]
- b. Critical Value [tex]\( = 1.65 \)[/tex]
- c. [tex]\( z = 2.23 \)[/tex]
- d. Reject [tex]\( H_0 \)[/tex]
- e. There is evidence to conclude that the average credit card debt has increased.
