Do students tend to improve their SAT mathematics (SAT-M) score the second time

Question

Do students tend to improve their SAT mathematics (SAT-M) score the second time they take the test? A random sample of four students who took the test twice received the following scores: Assuming that the change in SAT-M score (second score - first score) for the population of all students taking the test twice is Normally distributed with mean p, are we convinced that retaking the test improves scores? What is the P-value for a test of H_o: mu = muo versus Ha: mu notequalto mu_0? c. less than 0.01 a. more than 0.75 b. more than 0.10

Explanation / Answer

Let ud = u(2nd-1st)              

The differences are

-10
80
0
30

Formulating the null and alternative hypotheses,              
              
Ho:   ud   =   0  
Ha:   ud   =/   0  
At level of significance =    0.01          
As we can see, this is a    two   tailed test.      
              
Calculating the standard deviation of the differences (third column):              
              
s =    24.43296333          
              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    12.21648166          
              
Calculating the mean of the differences (third column):              
              
XD =    25          
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    2.04641571          
              
df = n - 1 =    3          
              
Hence, using p values,   using tables, we see that      
              
OPTION B: MORE THAN 0.10 [ANSWER]

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Note that using technology, the exact P value is p =    0.13322209, which fits the description.      
              

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