Questions: A credit card company claims that the mean credit card debt for individuals is greater than 4,900. You want to test this claim. You find that a random sample of 37 cardholders has a mean credit card balance of 5,178 and a standard deviation of 625. At α=0.10, can you support the claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (c) Find the standardized test statistic L. t= (Round to two decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. A. Reject H₀ because the test statistic is not in the rejection region. B. Fail to reject H₀ because the test statistic is not in the rejection region. C. Fail to reject H₀ because the test statistic is in the rejection region. D. Reject H₀ because the test statistic is in the rejection region.

A credit card company claims that the mean credit card debt for individuals is greater than 4,900. You want to test this claim. You find that a random sample of 37 cardholders has a mean credit card balance of 5,178 and a standard deviation of 625. At α=0.10, can you support the claim? Complete parts (a) through (d) below. Assume the population is normally distributed.

(c) Find the standardized test statistic L.
t= (Round to two decimal places as needed.)

(d) Decide whether to reject or fail to reject the null hypothesis.
A. Reject H₀ because the test statistic is not in the rejection region.
B. Fail to reject H₀ because the test statistic is not in the rejection region.
C. Fail to reject H₀ because the test statistic is in the rejection region.
D. Reject H₀ because the test statistic is in the rejection region.

Transcript text: A credit card company claims that the mean credit card debt for individuals is greater than $\$ 4,900$. You want to test this claim. You find that a random sample of 37 cardholders has a mean credit card balance of $\$ 5,178$ and a standard deviation of $\$ 625$. At $\alpha=0.10$, can you support the claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (c) Find the standardized test statistic L . t= $\square$ (Round to two decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. A. Reject $\mathrm{H}_{0}$ because the test statistic is not in the rejection region. B. Fail to reject $\mathrm{H}_{0}$ because the test statistic is not in the rejection region. C. Fail to reject $\mathrm{H}_{0}$ because the test statistic is in the rejection region. D. Reject $\mathrm{H}_{0}$ because the test statistic is in the rejection region.

Solution

Solution Steps

Step 1: Calculate the Standard Error

The standard error $ SE $ is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{625}{\sqrt{37}} \approx 102.75 \]

Step 2: Calculate the Test Statistic

The test statistic $ Z_{test} $ is calculated using the formula: \[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{5178 - 4900}{102.75} \approx 2.71 \]

Step 3: Determine the Critical Value

For a right-tailed test at $ \alpha = 0.10 $, the critical value is: \[ Z_{critical} \approx 1.2816 \]

Step 4: Make a Decision

Since the test statistic $ Z_{test} = 2.71 $ is greater than the critical value $ Z_{critical} = 1.2816 $, we reject the null hypothesis $ H_0 $.

Final Answer

The conclusion is that we can support the claim that the mean credit card debt for individuals is greater than $4,900. Thus, the answer is: \[ \boxed{D} \]

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