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A credit card company claims that the mean credit card debt for individuals is greater than $5,100\$5,100. You want to test this claim. You find that a random sample of 3333 cardholders has a mean credit card balance of $5,301\$5,301 and a standard deviation of $575\$575. At α=0.01\alpha=0.01, can you support the claim? Complete parts (a) through (e) below. Assume the population is normally distributed. (a) Write the claim mathematically and identily H0H_{0} and H2H_{2}. Which of the following correctly states H0H_{0} and H2H_{2} ?\newlineA.\newlineH0:μ=$5,100 H2:μ≠$5,100\begin{array}{l} H_{0}:\mu=\$5,100 \ H_{2}:\mu\neq\$5,100 \end{array}\newlineB.\newlineH0:μ≥$5,100 H2:μ<$5,100\begin{array}{l} H_{0}:\mu \geq \$5,100 \ H_{2}:\mu < \$5,100 \end{array}\newlineC.\newline$5,301\$5,30100\newlineD.\newline$5,301\$5,30111\newlineE.\newline$5,301\$5,30122\newlineF.\newline$5,301\$5,30133\newline(B) Find the critical value(s) and identify the rejection region(s). What is(are) the critical value(s), $5,301\$5,30144?\newline$5,301\$5,30155

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Q. A credit card company claims that the mean credit card debt for individuals is greater than $5,100\$5,100. You want to test this claim. You find that a random sample of 3333 cardholders has a mean credit card balance of $5,301\$5,301 and a standard deviation of $575\$575. At α=0.01\alpha=0.01, can you support the claim? Complete parts (a) through (e) below. Assume the population is normally distributed. (a) Write the claim mathematically and identily H0H_{0} and H2H_{2}. Which of the following correctly states H0H_{0} and H2H_{2} ?\newlineA.\newlineH0:μ=$5,100 H2:μ≠$5,100\begin{array}{l} H_{0}:\mu=\$5,100 \ H_{2}:\mu\neq\$5,100 \end{array}\newlineB.\newlineH0:μ≥$5,100 H2:μ<$5,100\begin{array}{l} H_{0}:\mu \geq \$5,100 \ H_{2}:\mu < \$5,100 \end{array}\newlineC.\newline$5,301\$5,30100\newlineD.\newline$5,301\$5,30111\newlineE.\newline$5,301\$5,30122\newlineF.\newline$5,301\$5,30133\newline(B) Find the critical value(s) and identify the rejection region(s). What is(are) the critical value(s), $5,301\$5,30144?\newline$5,301\$5,30155
  1. Write Claim and Identify Hypotheses: Write the claim mathematically and identify H0H_0 and HaH_a. The claim is that the mean credit card debt for individuals is greater than $5,100\$5,100. This is the alternative hypothesis (HaH_a). The null hypothesis (H0H_0) is the statement that the claim is not true, or that the mean is less than or equal to $5,100\$5,100. Therefore, the correct hypotheses are: H0:μ≤$5,100H_0: \mu \leq \$5,100 Ha:μ>$5,100H_a: \mu > \$5,100 The correct option is F.
  2. Find Critical Value and Rejection Region: Find the critical value(s) and identify the rejection region(s). Since we are dealing with a right-tailed test (because the claim is that the mean is greater than a certain value), we need to find the critical zz-value for α=0.01\alpha=0.01. Using a standard normal distribution table or a calculator, the critical zz-value for a right-tailed test at α=0.01\alpha=0.01 is approximately 2.332.33. The rejection region is z>2.33z > 2.33.
  3. Calculate Test Statistic: Calculate the test statistic.\newlineTo calculate the test statistic, we use the formula for the z-test:\newlinez=xˉ−μσ/nz = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\newlinewhere xˉ\bar{x} is the sample mean, μ\mu is the population mean under the null hypothesis, σ\sigma is the population standard deviation, and nn is the sample size.\newlinePlugging in the values, we get:\newlinez=5301−5100575/33z = \frac{5301 - 5100}{575/\sqrt{33}}\newlinez≈201575/33z \approx \frac{201}{575/\sqrt{33}}\newlinez≈201100.22z \approx \frac{201}{100.22}\newlinez≈2.01z \approx 2.01
  4. Make Decision: Make a decision.\newlineSince the calculated zz-value of 2.012.01 is less than the critical zz-value of 2.332.33, we do not reject the null hypothesis.\newlineThis means that we do not have enough evidence to support the claim that the mean credit card debt for individuals is greater than $5,100\$5,100 at the 0.010.01 significance level.

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