A gambler tosses a fair die 20 times and computes the sum X

A gambler tosses a fair die 20 times and computes the sum, X, of all the tosses. Use the central limit theorem to approximate the probability that X is between 70 and 100.

Solution

Consider:

Thus,  
  
E(x) = Expected value = mean =    3.5
Var(x) = E(x^2) - E(x)^2 =    2.916666667
s(x) = sqrt [Var(x)] =    1.707825128

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound = 70/20 =   3.5      
x2 = upper bound = 100/20 =   5      
u = mean =    3.5      
n = sample size =    20      
s = standard deviation =    1.707825128      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    0      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    3.927922023      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.5      
P(z < z2) =    0.999957159      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.499957159 [ANSWER]

x	P(x)	x P(x)	x^2 P(x)
1	0.166667	0.166667	0.166667
2	0.166667	0.333333	0.666667
3	0.166667	0.5	1.5
4	0.166667	0.666667	2.666667
5	0.166667	0.833333	4.166667
6	0.166667	1	6