If a sample size of 50 is drawn from a normal population tha

If a sample size of 50 is drawn from a normal population that has mean 275 and variance s1, what is the probability that the sample mean differ from population mean try greater that 1?

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound = 275 - 1 =   274      
x2 = upper bound = 275 + 1 =   276      
u = mean =    275      
n = sample size =    50      
s = standard deviation =    9      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -0.785674201      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    0.785674201      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.216029191      
P(z < z2) =    0.783970809      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.567941619      

Thus, those outside this interval is the complement =    0.432058381   [ANSWER]