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

Question

5. Hypothesis tests about a population mean, population standard deviation known Aa Aa 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) is713, 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: O Ho:

Explanation / Answer

Q5.
Given that,
population mean(u)=731
standard deviation, =76
sample mean, x =713
number (n)=64
null, Ho: >=731
alternate, H1: <731
level of significance, = 0.05
from standard normal table,left tailed z /2 =1.645
since our test is left-tailed
reject Ho, if zo < -1.645
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 713-731/(76/sqrt(64)
zo = -1.89
| zo | = 1.89
critical value
the value of |z | at los 5% is 1.645
we got |zo| =1.89 & | z | = 1.645
make decision
hence value of | zo | > | z | and here we reject Ho
p-value : left tail - ha : ( p < -1.89 ) = 0.03
hence value of p0.05 > 0.03, here we reject Ho
ANSWERS
---------------
null, Ho: >=731
alternate, H1: <731
test statistic: -1.89
reject Ho, if zo <= -1.645
p-value: 0.03
critical value: -1.645
decision: reject Ho

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