In a sample of 25 Brooklyn undergraduates suppose that 10 re

In a sample of 25 Brooklyn undergraduates, suppose that 10 receive no financial aid, 8 borrow $2,000 per year, 6 borrow $4,000 per year, and 1 borrows $5,000 per year.
a. Test the null hypothesis that the mean annual financial aid burden of undergraduates is equal to $2,000 at the 0.05 level of significance. (Show all steps in your work and clearly state your conclusion.) Please use critical value approach.

b. Test the claim that the mean annual financial aid burden of undergraduates is less than $1,500. Use the 0.01 level of significance. (Show all steps in your work and clearly state your conclusion.) Please use P-value approach.

Solution

With the data set

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2000
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2000
4000
4000
4000
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4000
4000
5000

The mean and standard deviations are

X = 1800
s = 1732.050808

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   2000  
Ha:    u   =/   2000  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical t,              
df = n - 1 =    24          
tcrit =    +/-   2.063898562      
              
Getting the test statistic, as              
              
X = sample mean =    1800          
uo = hypothesized mean =    2000          
n = sample size =    25          
s = standard deviation =    1732.050808          
              
Thus, t = (X - uo) * sqrt(n) / s =    -0.577350269          
              
Comparing t and tcrit, FAIL TO REJECT THE NULL HYPOTHESIS.      
  
Thus, there is no significant evidence that the   mean annual financial aid burden of undergraduates is not equal to $2,000.

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b)

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   1500  
Ha:    u   <   1500  
              
As we can see, this is a    left   tailed test.      
                          
Getting the test statistic, as              
              
X = sample mean =    1800          
uo = hypothesized mean =    1500          
n = sample size =    25          
s = standard deviation =    1732.050808          
              
Thus, t = (X - uo) * sqrt(n) / s =    0.866025404          
              
Also, the p value is, as df = 24,
              
p = 0.802474241
              
As p > 0.01, FAIL TO REJECT THE NULL HYPOTHESIS.  

Thus, there is   no significant evidence that the   mean annual financial aid burden of undergraduates is less than $1500.