An online clothing company knows that 36 of i ts customers t

An online clothing company knows that 36% of i ts customers that buy a certain style of coat choos e email. Of the next 85 customers, what is the probability that bet ween 25 and 35 customers (inclusive) will buy an em ail coat? Use a normal approximation to the binomial.

Solution

Here,

Mean = n p = 85*0.36 = 30.6

Also,

standard deviation = sqrt(n p (1-p)) = sqrt(85*0.36*(1-0.36)) = 4.42538134

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    24.5      
x2 = upper bound =    35.5      
u = mean =    30.6      
          
s = standard deviation =    4.42538134      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.378412284      
z2 = upper z score = (x2 - u) / s =    1.107249212      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.084038016      
P(z < z2) =    0.865906902      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.781868885   [ANSWER]