Questions: A credit card company claims that the mean credit card debt for individuals is greater than 4,800. You want to test this claim. You find that a random sample of 27 cardholders has a mean credit card balance of 5,060 and a standard deviation of 600. At α=0.05, can you support the claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (a) Write the claim mathematically and identify H0 and Ha. Which of the following correctly states H0 and Ha ? A. H0: μ= 4,800 B. H0: μ= 4,800 Ha: μ ≠ 4,800 D. H0: μ ≥ 4,800 E. H0: μ ≤ 4,800 Ha: μ< 4,800 Ha: μ> 4,800 F. H0: μ> 4,800 Ha: μ ≤ 4,800 (b) Find the critical value and identify the rejection region. What is the critical value, t0 ? t0= (Use a comma to separate answers as needed. Round to three decimal places as needed.) Determine the rejection region. Select the correct choice below and fill in the answer box within your choice. (Round to three decimal places as needed.) A. <t< B. t< C. t> D. t< and t>

A credit card company claims that the mean credit card debt for individuals is greater than 4,800. You want to test this claim. You find that a random sample of 27 cardholders has a mean credit card balance of 5,060 and a standard deviation of 600. At α=0.05, can you support the claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (a) Write the claim mathematically and identify H0 and Ha. Which of the following correctly states H0 and Ha ? A. H0: μ= 4,800 B. H0: μ= 4,800 Ha: μ ≠ 4,800 D. H0: μ ≥ 4,800 E. H0: μ ≤ 4,800 Ha: μ< 4,800 Ha: μ> 4,800 F. H0: μ> 4,800 Ha: μ ≤ 4,800 (b) Find the critical value and identify the rejection region. What is the critical value, t0 ? t0= (Use a comma to separate answers as needed. Round to three decimal places as needed.) Determine the rejection region. Select the correct choice below and fill in the answer box within your choice. (Round to three decimal places as needed.) A. <t< B. t< C. t> D. t< and t>
Transcript text: A credit card company claims that the mean credit card debt for individuals is greater than $\$ 4,800$. You want to test this claim. You find that a random sample of 27 cardholders has a mean credit card balance of $\$ 5,060$ and a standard deviation of $\$ 600$. At $\alpha=0.05$, can you support the claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (a) Write the claim mathematically and identify $\mathrm{H}_{0}$ and $\mathrm{H}_{a}$. Which of the following correctly states $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$ ? A. $H_{0}: \mu=\$ 4,800$ B. $H_{0}: \mu=\$ 4,800$ $H_{a}: \mu \neq \$ 4,800$ D. $H_{0}: \mu \geq \$ 4,800$ E. $\mathrm{H}_{0}: \mu \leq \$ 4,800$ $H_{a}: \mu<\$ 4,800$ $H_{a}: \mu>\$ 4,800$ F. $H_{0}: \mu>\$ 4,800$ $H_{a}: \mu \leq \$ 4,800$ (b) Find the critical value and identify the rejection region. What is the critical value, $\mathrm{t}_{0}$ ? \[ \mathrm{t}_{0}=\square \] (Use a comma to separate answers as needed. Round to three decimal places as needed.) Determine the rejection region. Select the correct choice below and fill in the answer box within your choice. (Round to three decimal places as needed.) A. $\square$ $$ $\square$ D. $t<$ $\square$ and $t>$ $\square$
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Solution

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Solution Steps

Step 1: State the Hypotheses

The claim is that the mean credit card debt for individuals is greater than \$4,800. We can express the null and alternative hypotheses as follows: \[ H_0: \mu \leq 4800 \quad \text{(Null Hypothesis)} \] \[ H_a: \mu > 4800 \quad \text{(Alternative Hypothesis)} \]

Step 2: Calculate the Standard Error

The standard error (SE) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{600}{\sqrt{27}} \approx 115.47 \]

Step 3: Calculate the Test Statistic

The test statistic \(t_{\text{test}}\) is calculated using the formula: \[ t_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{5060 - 4800}{115.47} \approx 2.252 \]

Step 4: Calculate the P-value

For a right-tailed test, the P-value is calculated as: \[ P = 1 - T(z) \approx 0.017 \]

Step 5: Determine the Critical Value and Rejection Region

For a right-tailed test at \(\alpha = 0.05\), the critical value \(t_0\) is: \[ t_0 \approx 1.6449 \] The rejection region is defined as: \[ t > 1.6449 \]

Final Answer

Since the test statistic \(t_{\text{test}} \approx 2.252\) is greater than the critical value \(t_0 \approx 1.6449\) and the P-value \(0.017\) is less than \(\alpha = 0.05\), we reject the null hypothesis. Thus, we have sufficient evidence to support the claim that the mean credit card debt for individuals is greater than \$4,800.

\(\boxed{\text{The answer is E: } H_0: \mu \leq 4800, H_a: \mu > 4800}\)

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