Question

2. Hypothesis tests about a population mean, population standard deviation known

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 64 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 = 713. 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, Haa: µ < 731

H₀: x̄x̄ ≥ 731, Haa: x̄x̄ < 731

H₀: µ ≤ 731, Haa: µ > 731

H₀: µ < 731, Haa: µ ≥ 731

If the null hypothesis is true as an equality, the sampling distribution of x̄x̄ is approximated by ________ distribution with _________ and a standard deviation of .

The value of the standardized test statistic is_______.

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

Normal Distribution

Mean = 730

Standard Deviation = 8.5

710720730740750x̄z-2-1012z

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 H₀ if z ≤ -1.645

Reject H₀ if z ≤ –1.96 or z ≥ 1.96

Reject Haa if z ≤ -1.645

Reject H₀ if z ≥ -1.89

The p-value is .

Using the critical value approach, the null hypothesis is ________ , because__________ . Using the p-value approach, the null hypothesis is ________ , because ______ . Therefore, you_______ conclude that banks have loosened their standards for issuing credit cards since 2002.

Answer 1

Solution: Since we are to check iwhether banks have loosened their standards for issuing credit cards since 2002, we have to check whether the FICO scores have decreased in the data of past 6 months. For that, we construct the null and alternative hypotheses as:

H0: mu >= 731 vs Ha: mu < 731 where mu is the mean FICO score of the population.

The test statistic is Z= (xbar-mu0)/(sigma/sqrt(n)) ; where xbar = sample mean, mu0 = the hypothesized value of the population mean, n = sample size, s = population standard deviation, sqrt refers to the square root function.

Under H0, Z ~ N(0,1)

We reject H0 if Z(observed) < -tau(alpha), where tau(alpha) is the upper alpha point of the standard normal - distribution.   

Or we reject H0 if p-value for the obtained statistics is less than alpha where alpha = level of significance.

Thus, According to the critical value approach, we reject the null hypothesis when

z ≤ -1.645 since -tau(alpha) = -1.645 (Obtained from the probability table of Standard normal distribution)

Here n= 64, sigma = 76, mu = 731, xbar = 713

So, Here, Z(observed) =   -1.894737

Thus, Using the critical value approach the null hypothesis is rejected because Z(observed) < - tau(alpha).

Using the p-value approach, the null hypothesis is rejected because p-value is 0.029, which is less than 0.05(alpha).

Therefore, we conclude at a 5% level of significance on the basis of the given sample that there is enough evidence to support the claim that the banks have loosened their standards for issuing credit cards since 2002.