1 For a random variable that is normally distributed with 1

1. For a random variable that is normally distributed, with = 113.75 and = 26.7754, the probability that a simple random sample of 35 items will produce a mean that is between 119 and 120.1315 is equal to

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    119      
x2 = upper bound =    120.1315      
u = mean =    113.75      
n = sample size =    43      
s = standard deviation =    26.7754      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    1.285753051      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    1.562863447      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.90073537      
P(z < z2) =    0.940957642      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.040222272   [ANSWER]