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

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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 his 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 tightened 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 747. Assume that the standard deviation of the population of FICO scores for credit cards issued during the past 6 months is known to be -76, the standard deviation from the NBER study. Let 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 Ho: 731, Ha: > 731 Ho: > 731, Hai 731 Ho: 731, Hai 731 If the null hypothesis is true as an equality, the sampling distribution of is approximated by distribution with and a standard deviation of

Explanation / Answer

1.

Formulating the null and alternative hypotheses,              
              
Ho:   u   <=   731  
Ha:    u   >   731   [OPTION A]

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2.

If the null hypothesis is true as an equality, the sampling distirbution of xbar is approximated by a [[normal distribution]] with mean [[731]] and a standard deviation of [[7.6]].

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3.
              
As we can see, this is a    right   tailed test.  
Getting the test statistic, as              
              
X = sample mean =    747          
uo = hypothesized mean =    731          
n = sample size =    100          
s = standard deviation =    76          
              
Thus, z = (X - uo) * sqrt(n) / s =    2.105263158 = 2.11   [ANSWER, STANDARDIZED TEST STATISTIC]

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4.  
              
              
Also, the p value is              
              
p =    0.017634204 [exact] --> 0.0174 [ANSWER]

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