Credit ScoresA Fair Isaac Corporation (FICO) 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 FICO score over 700 considered to be a quality credit risk. According to Fair Isaac Corporation, the mean FICO score is 703.5. 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 40 high-income individuals and found the sample mean credit score to be 714.2 with a standard deviation of 83.2. Conduct the appropriate test to determine if high-income individuals have higher FICO scores at the α = 0.05 level of significance.

Verified step by step guidance

1. Identify the null and alternative hypotheses. Since we want to test if high-income individuals have higher FICO scores, set the null hypothesis as \(H_0: \mu = 703.5\) (mean FICO score is equal to the population mean) and the alternative hypothesis as \(H_a: \mu > 703.5\) (mean FICO score is greater than the population mean).
2. Determine the significance level \(\alpha = 0.05\) and note that this is a one-tailed test because the alternative hypothesis is testing for a mean greater than the population mean.
3. Calculate the test statistic using the formula for a one-sample t-test: \(\displaystyle t = \frac{\overline{x} - \mu_0}{s / \sqrt{n}}\) where \(\overline{x} = 714.2\) is the sample mean, \(\mu_0 = 703.5\) is the population mean, \(s = 83.2\) is the sample standard deviation, and \(n = 40\) is the sample size.
4. Find the critical t-value from the t-distribution table with \(df = n - 1 = 39\) degrees of freedom at the \(\alpha = 0.05\) significance level for a one-tailed test.
5. Compare the calculated t-statistic to the critical t-value: if the t-statistic is greater than the critical value, reject the null hypothesis, concluding that high-income individuals have significantly higher FICO scores; otherwise, do not reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this context, it involves testing if the mean FICO score of high-income individuals is significantly greater than the known population mean. The test compares sample data against a significance level (α) to make an inference.

One-Sample t-Test

A one-sample t-test is used to compare the sample mean to a known population mean when the population standard deviation is unknown and the sample size is relatively small. It calculates a t-statistic based on the sample mean, sample standard deviation, and sample size to determine if the observed difference is statistically significant.

Significance Level and p-Value

The significance level (α) is the threshold for rejecting the null hypothesis, commonly set at 0.05. The p-value measures the probability of observing the sample data, or something more extreme, if the null hypothesis is true. If the p-value is less than α, the null hypothesis is rejected, indicating a statistically significant result.