From past experience the cognition scores on a certain test

From past experience, the cognition scores on a certain test are known to have a mean of 50 and = 14. You have tested a random sample of 40 subjects who have a history of depression and found a sample mean of 48.

A. Is the significant evidence that the mean score in this population differs from 50? Use alpha = 5% and assume that the standard deviation is still 14. In your conclusion, identify the relevant population.

B. Is it necessary to assume the data come from a normal distribution? Why or why not?

Solution

A.

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   50  
Ha:    u   =/   50  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical z, as alpha =    0.05   ,      
alpha/2 =    0.025          
zcrit =    +/-   1.959963985      
              
Getting the test statistic, as              
              
X = sample mean =    48          
uo = hypothesized mean =    50          
n = sample size =    40          
s = standard deviation =    14          
              
Thus, z = (X - uo) * sqrt(n) / s =    -0.903507903          
              
Also, the p value is              
              
p =    0.366256396          
              
As |z| < 1.95996, and P > 0.05, we   FAIL TO REJECT THE NULL HYPOTHESIS.   [ANSWER]

Thus, there is no significant evidence that the true population mean of cognition scores on the test differs from 50. [CONCLUSION]

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B.

No, because the sample size n = 40 is big enough so that the sampling distribution of the mean is approimately normal already, by central limit theorem.