A normal distribution of scores in population has a mean of

A normal distribution of scores in population has a mean of µ = 100 with = 20. A. What is the probability of randomly selecting a score greater than X = 110 from this population? B. If a sample of n = 25 scores is randomly selected from this population, what is the probability that the sample mean will be greater than M = 110?

Solution

a)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    100      
u = mean =    20      
          
s = standard deviation =    110      
          
Thus,          
          
z = (x - u) / s =    0.727272727      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.727272727   ) =    0.233529451 [answer]

b)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    110      
u = mean =    100      
n = sample size =    25      
s = standard deviation =    20      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    2.5      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.5   ) =    0.006209665 [answer]