The recent default rate on all student loans is 71 percent I

The recent default rate on all student loans is 7.1 percent. In a recent random sample of 282 loans at private universities, there were 15 defaults.

a.)

Does this sample show sufficient evidence that the private university loan default rate is below the rate for all universities, using a left-tailed test at = .01?

Choose the appropriate hypothesis.

Calculate the z-score for the sample data using a left-tailed test at = .01? (Round your answer to 3 decimal places.)

Should the null hypothesis be rejected?

b.) Calculate the p-value. (Round your answer to 4 decimal places.)

c.) Is the assumption of normality justified?

Solution

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.071
Ha:   p   <   0.071 [PART A]

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As we see, the hypothesized po =   0.071      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.053191489      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.015293697      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -1.164434657      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.2442   [PART B]

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significance level =    0.01
      
Comparing p and the significance value, we   FAIL TO REJECT THE NULL HYPOTHESIS.      

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If you assume normality of np > 10 and n(1 - p) > 10, then yes it does.