Question

Lenders tighten or loosen their standards for issuing credit as economic conditions change. One of the criteria lenders use to evaluate the creditworthiness of a potential borrower is her credit risk score, usually a FICO score. FICO scores range from 300 to 850. A consumer with a high FICO score is perceived to be a low credit risk to the lender and is more likely to be extended credit than a consumer with a low score.

A credit card represents a line of credit, because the credit card holder obtains a loan whenever the card is used to pay for a purchase. A study of credit card accounts opened in 2002 found a mean FICO score for the credit card holder (at the time the card was issued) of 731 and a standard deviation of 76. [Source: Sumit Agarwal, John C. Driscoll, Xavier Gabaix, and David Laibson, “Learning in the Credit Card Market,” Working Paper 13822, National Bureau of Economic Research (NBER), February 2008.]

You conduct a hypothesis test to determine whether banks have loosened their standards for issuing credit cards since 2002. You collect a random sample of 100 credit cards issued during the past 6 months. The sample mean FICO score of the credit card holders (at the time their cards were issued) is x̄x̄ = 715. Assume that the standard deviation of the population of FICO scores for credit cards issued during the past 6 months is known to be σ = 76, the standard deviation from the NBER study.

Let µ equal the true population mean FICO score for consumers issued credit cards in the past 6 months. You should formulate the null and alternative hypotheses as:

H₀: µ = 731, H11 : µ < 731

H₀: µ = 731, H11 : µ > 731

H₀: x̄x̄ = 731, H11 : x̄x̄ < 731

H₀: µ < 731, H11 : µ = 731

If the null hypothesis is true as an equality, the sampling distribution of x̄x̄ is approximated by selector 1   

• a t
• a standard normal
• a normal
• a binomial
• an unknown

distribution with selector 2   

• a mean of 0
• an unknown mean
• a mean of 715
• a mean of 731

and a standard deviation of selector 3   

• 76
• 1.0
• unknown
• 7.6

.
The value of the standardized test statistic is selector 1   

• z = 16
• t = –2.11
• z = –2.11
• z = 715

.

Use the Distributions tool to help you answer the questions that follow.

Normal Distribution

Mean = 730

Standard Deviation = 8.5

You conduct the hypothesis test using a significance level of α = 0.05. Use the tool to develop the rejection region for your test. According to the critical value approach, when do you reject the null hypothesis?

Reject H1 if z ≤ –1.645

Reject H₀ if z ≥ –2.11

Reject H₀ if z ≤ –1.645

Reject H₀ if z ≤ –1.96 or z ≥ 1.96

The p-value is selector 1   

• 16
• 0.9826
• –2.11
• 0.0174

.
Using the critical value approach, the null hypothesis is selector 1   

• not rejected
• rejected

, because selector 2   

• –2.11 < –1.645
• 7.6 > –1.645
• 0.0174 < 0.05
• 715 < 731

. Using the p-value approach, the null hypothesis is selector 3   

• not rejected
• rejected

, because selector 4   

• 0.0174 < 0.05
• 0.9826 > 0.05
• 0.0143 < 0.05
• –2.11 < –1.645

. Therefore, you selector 5   

• cannot
• can

conclude that banks have loosened their standards for issuing credit cards since 2002.

Answer 1