Question

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 35 cardholders has a mean credit card balance of $5,154 and a standard deviation of $550. at \\( \alpha = 0.10 \\), can you support the claim? complete parts (a) through (e) below. assume the population is normally distributed.
(a) decide whether to reject or fail to reject the null hypothesis.
a. fail to reject \\( h _ { 0 } \\) because the test statistic is not in the rejection region.
b. reject \\( h _ { 0 } \\) because the test statistic is in the rejection region.
c. reject \\( h _ { 0 } \\) because the test statistic is not in the rejection region.
d. fail to reject \\( h _ { 0 } \\) because the test statistic is in the rejection region.
(e) interpret the decision in the context of the original claim.
a. at the 10% level of significance, there is sufficient evidence to support the claim that the mean credit card debt is greater than $4,900.
b. at the 10% level of significance, there is sufficient evidence to support the claim that the mean credit card debt is less than $4,900.
c. at the 10% level of significance, there is not sufficient evidence to support the claim that the mean credit card debt is less than $4,900.
d. at the 10% level of significance, there is not sufficient evidence to support the claim that the mean credit card debt is greater than $4,900.

Explanation:

• For part (a), since we are testing a claim about a population mean and the population is normally distributed (even though the sample size \(n = 35\geq30\) which also gives us the right to use the z - test due to the Central Limit Theorem), and we assume that the null hypothesis \(H_0:\mu\leq4900\) and the alternative hypothesis \(H_1:\mu > 4900\). The test statistic \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\bar{x} = 5154\), \(\mu=4900\), \(\sigma = 550\), \(n = 35\). The critical value for a right - tailed test with \(\alpha=0.10\) is \(z_{\alpha}=1.28\). If the calculated \(z\) - statistic is greater than \(z_{\alpha}\), we reject \(H_0\).
• For part (e), when we reject \(H_0\) (the null hypothesis that \(\mu\leq4900\)), it means we support the alternative hypothesis. In the context of hypothesis testing, rejecting the null hypothesis in favor of the alternative hypothesis (which is the claim in this case) implies that there is sufficient evidence to support the claim.

Answer:

(a) B. Reject \(H_0\) because the test statistic is in the rejection region.
(e) A. At the \(10\%\) level of significance, there is sufficient evidence to support the claim that the mean credit card debt is greater than \(\$4,900\).