The average credit card debt for a recent year was [tex]$\$[/tex] 8624[tex]$. Five years earlier, the average credit card debt was $[/tex]\[tex]$ 8021$[/tex]. Assume sample sizes of 32 were used and the population standard deviations of both samples were [tex]$\$[/tex] 1083[tex]$. Is there evidence to conclude that the average credit card debt has increased? Use $[/tex]\alpha=0.05[tex]$.

a. State the hypotheses.
b. Find the critical value.
c. Compute the test statistic.
d. Make the decision.
e. Summarize the results.

A)
a. $[/tex]H_0: \mu_1=\mu_2 ; H_1: \mu_1<\mu_2[tex]$
b. C.V. $[/tex]=1.96[tex]$
c. $[/tex]z=2.84[tex]$
d. Do not reject
e. There is not enough evidence to conclude that the average credit card debt has increased.

B)
a. $[/tex]H_0: \mu_1=\mu_2 ; H_1: \mu_1<\mu_2[tex]$
b. C.V. $[/tex]=1.65[tex]$
c. $[/tex]z=2.23[tex]$
d. Reject
e. There is evidence to conclude that the average credit card debt has increased.

C)
a. $[/tex]H_0: \mu_1=\mu_2 ; H_1: \mu_1<\mu_2[tex]$
b. C.V. $[/tex]=1.65[tex]$
c. $[/tex]z=2.23[tex]$
d. Do not reject
e. There is not enough evidence to conclude that the average credit card debt has increased.

D)
a. $[/tex]H_0: \mu_1=\mu_2 ; H_1: \mu_1<\mu_2[tex]$
b. C.V. $[/tex]=1.96[tex]$
c. $[/tex]z=2.84$
d. Reject
e. There is evidence to conclude that the average credit card debt has increased.



Answer :

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.