When consumers apply for credit, their credit is rated using...

When consumers apply for credit, their credit is rated using FICO (Fair, Isaac, and Company) scores. A random sample of credit ratings is obtained, and the FICO scores are summarized with these statistics: $n=18, \bar{x}=660.3, s=95.9$ Use a 0.05 significance level to test the claim that these credit ratings are from a population with a mean that is equal to $700 .$ If the Bank of Newport requires a credit rating of 700 or higher for a car loan, do the results indicate that everyone will be eligible for a car loan? Why or why not?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter based on sample data. It involves setting up a null hypothesis that represents a baseline claim (e.g., the population mean is equal to a specific value) and an alternative hypothesis that represents a challenge to that baseline. The procedure then tests whether the sample evidence is strong enough to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis (H0) typically states that there is no difference or effect (in this case, that the population mean is equal to 700), while the alternative hypothesis (H1) suggests that the population mean is not 700. This formulation is crucial as it establishes the framework for the statistical test and determines the direction and criteria for rejection of the null hypothesis.

t-Test and t-Distribution

When the sample size is small and the population standard deviation is unknown, the t-test is used to compare the sample mean to a population mean. The t-distribution, which has heavier tails than the normal distribution, is used to account for the added variability in estimation. This concept is fundamental when working with sample statistics to infer about the population mean.

Significance Level

The significance level, often denoted by ? (in this case, 0.05), is the threshold probability used for deciding whether to reject the null hypothesis. It represents the maximum probability of committing a Type I error (rejecting the null hypothesis when it is actually true) that the researcher is willing to accept.

Statistical vs. Practical Significance

Even if a statistical test leads to rejecting the null hypothesis, this does not automatically imply practical implications for every individual in the population. The concept addresses the difference between statistically significant results and their real-world impact. In the context of credit ratings, even if the mean is statistically different from 700, variability around the mean means that many individual scores may still meet the requirement for a car loan.

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