Question

A​ government-sponsored organization that was established in 1938 after the Depression to provide local banks with money from the federal government to be used for residential mortgages in an effort to increase homeownership rates. As lending standards have tightened during the most recent housing market​ crisis, credit scores for borrowers of approved mortgages have increased. In​ 2012, the average credit score for loans that were purchased by the company was 782. A random sample of 25 mortgages recently purchased by the company was selected and it was found that the average credit score was 789 with a sample standard deviation of 25. Complete parts​ (a) through​ (c) below. a. Using alphaequals0.05​, is there enough evidence from this sample to conclude that the average credit score increased as the lending standards have tightened​? Determine the null and alternative hypotheses. Choose the correct answer below. A.Upper H 0​: mu not equals 782 and Upper H 1​: mu equals 782 B.Upper H 0​: mu greater than or equals 782 and Upper H 1​: mu less than 782 C.Upper H 0​: mu less than or equals 782 and Upper H 1​: mu greater than 782 D.Upper H 0​: mu equals 782 and Upper H 1​: mu not equals 782 Determine the critical​ value(s). The critical​ value(s) is(are) nothing. ​(Round to three decimal places as needed. Use a comma to separate answers as​ needed.) Determine the test​ statistic, t Subscript x overbar. t Subscript x overbarequals nothing ​(Round to two decimal places as​ needed.) What conclusion should be​ drawn? A.Do not reject Upper H 0. There is not sufficient evidence to conclude that the average credit score increased as the lending standards have tightened. B.Do not reject Upper H 0. There is sufficient evidence to conclude that the average credit score increased as the lending standards have tightened. C.Reject Upper H 0. There is sufficient evidence to conclude that the average credit score increased as the lending standards have tightened. D.Reject Upper H 0. There is not sufficient evidence to conclude that the average credit score increased as the lending standards have tightened. b. Use technology to determine the​ p-value for this test. ​p-valueequals nothing ​(Round to three decimal places as​ needed.) c. What assumptions need to be made to perform this​ analysis? A.That there was less than 30 users sampled. B.Because the sample size is less than​ 30, the population needs to be normally distributed. C.That the median was less than the mean value. D.No assumptions were needed to preform the analysis. Click to select your answer(s).

Answer 1

a)

Ho :   µ =   782          
Ha :   µ >   782       (Right tail test)  

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Level of Significance ,    α =    0.05

degree of freedom=    DF=n-1=    24                     
                          
critical t value, t* =        1.711 [Excel formula =t.inv(α,df) ]              

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sample std dev ,      s =     25.0000                    
Sample Size ,    n =     25                    
Sample Mean,      x̅ =    789.0000                    
                            
                            
Standard Error , SE =    s/√n =    25.0000    / √     25    =    5.0000        
t-test statistic= (x̅ - µ )/SE = (    789.000    -    782    ) /     5.000    =    1.400

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decision: test stat< critical value, so, do not reject Ho

Do not reject Upper H 0. There is not sufficient evidence to conclude that the average credit score increased as the lending standards have tightened.

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b)

p-Value   =   0.087 [Excel formula =t.dist(t-stat,df) ]

c)

Because the sample size is less than​ 30, the population needs to be normally distributed.