Thompson and Thompson is a steel bolts manufacturing company

Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 139 millimeters, and a variance of 49. If a random sample of 34 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 1.8 millimeters? Round your answer to four decimal places.

Solution

We can get the lower and upper bounds by adding/subtracting 1.8 from the mean.

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    137.2      
x2 = upper bound =    140.8      
u = mean =    139      
n = sample size =    34      
s = standard deviation =    7      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -1.49938763      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    1.49938763      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.06688655      
P(z < z2) =    0.93311345      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.866226899   [ANSWER]