Question

A credit score is used by credit agencies​ (such as mortgage companies and​ banks) to assess the creditworthiness of individuals. Values range from 300 to​ 850, with a credit score over 700 considered to be a quality credit risk. According to a​ survey, the mean credit score is 703.1. A credit analyst wondered whether​ high-income individuals​ (incomes in excess of​ $100,000 per​ year) had higher credit scores. He obtained a random sample of 35 ​high-income individuals and found the sample mean credit score to be 716.6 with a standard deviation of 80.1. Conduct the appropriate test to determine if​ high-income individuals have higher credit scores at the alphaequals0.05 level of significance.

Answer 1

To determine if high-income individuals have higher credit scores, we can conduct a hypothesis test at the alpha equals 0.05 level of significance. The null hypothesis states that high-income individuals have the same credit scores as the general population, while the alternative hypothesis suggests that high-income individuals have higher credit scores. By calculating the test statistic and comparing it to the critical value, we can make a decision on whether to reject or fail to reject the null hypothesis. In this case, we reject the null hypothesis, indicating that high-income individuals do have higher credit scores.

Step-by-step explanation:

To test whether high-income individuals have higher credit scores at the alpha equals 0.05 level of significance, we can conduct a hypothesis test using the t-distribution. Here are the steps:

1. Step 1: State the hypotheses:
   Null hypothesis (H0): High-income individuals have the same credit scores as the general population (mean equals 703.1).
   Alternative hypothesis (Ha): High-income individuals have higher credit scores than the general population (mean is greater than 703.1).
2. Step 2: Select the level of significance (alpha) = 0.05.
3. Step 3: Compute the test statistic:
   The test statistic is calculated using the formula: t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size)).
   In this case:
   t = (716.6 - 703.1) / (80.1 / sqrt(35)) = 2.78 (rounded to two decimal places).
4. Step 4: Determine the critical value:
   Since it is a one-tailed test with alpha equals 0.05, the critical value is found by looking up t-distribution table with 34 degrees of freedom and a one-tailed alpha equals 0.05.
   The critical value is approximately 1.689 (rounded to three decimal places).
5. Step 5: Make a decision:
   If the test statistic (2.78) is greater than the critical value (1.689), we reject the null hypothesis.
   Otherwise, we fail to reject the null hypothesis.
6. Step 6: Interpret the result:
   Since the test statistic (2.78) is greater than the critical value (1.689), we reject the null hypothesis.
   Therefore, there is evidence to suggest that high-income individuals have higher credit scores than the general population at the alpha equals 0.05 level of significance.