Need help with B C A simple random sample of size n 40 is

Need help with B & C

A simple random sample of size n = 40 is obtained from a population with mu = 64 and sigma = 19. What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of Assuming the normal model can be used, determine P(

Solution

b)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    67.1      
u = mean =    64      
n = sample size =    40      
s = standard deviation =    19      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    1.031901131      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   1.031901131   ) =    0.848940782 [answer]

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c)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    65.6      
u = mean =    64      
n = sample size =    40      
s = standard deviation =    19      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.532594132      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.532594132   ) =    0.297157282 [answer]