Assume a binomial probability distribution with n40 and p05

. Assume a binomial probability distribution with n=40 and p=0.55. Compute the following:

a. The mean and standard deviation of the random variable.

b. The probability that x is between 15 and 25, inclusive. (Hint: use the normal approximation to the binomial. First check to see if the conditions are met).

Solution

a)

Mean = n p = 40*0.55 = 22 [ANSWER]

Standard deviation = sqrt(n p (1-p)) = sqrt(40*0.55*(1-0.55)) = 3.146426545 [ANSWER]

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b)

As np = 22, nq = 18, then the conditions are met (both are greater than 10).

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    14.5      
x2 = upper bound =    25.5      
u = mean =    22      
          
s = standard deviation =    3.146426545      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -2.383656473      
z2 = upper z score = (x2 - u) / s =    1.112373021      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.008570798      
P(z < z2) =    0.8670111      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.858440302   [ANSWER]