Suppose that Sn is a binomial random variable with probabili

Suppose that Sn is a binomial random variable with probability p = 1/3. Take n = 18 and use the normal approximation to approximate the following probabilities. First calculate ES18 = and V(S18) = . P(S18 LE 7) P(S18 LE 7.5) P(5 LE S18 LE 7) Take n = 72 and use the normal approximation to find the following probabilities. First calculate ES72 = and V(S72) = . P(S72 LE 28) P(18 LE S72 LE 30) P(S72 = 24)

Solution

2.

a)

ES18 = u = mean = np =    6   [ANSWER]  
          
V(S18) = np(1-p) =    4   [ANSWER]

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We first get the z score for the critical value:          
          
x = critical value =    7      
u = mean = np =    6      
          
s = standard deviation = sqrt(np(1-p)) =    2      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    0.5      
          
Thus, the left tailed area is          
          
P(z <   0.5   ) =    0.691462461 [ANSWER]

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We first get the z score for the critical value:          
          
x = critical value =    7.5      
u = mean = np =    6      
          
s = standard deviation = sqrt(np(1-p)) =    2      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    0.75      
          
Thus, the left tailed area is          
          
P(z <   0.75   ) =    0.773372648 [ANSWER]

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We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    5      
x2 = upper bound =    7      
u = mean = np =    6      
          
s = standard deviation = sqrt(np(1-p)) =    2      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.5      
z2 = upper z score = (x2 - u) / s =    0.5      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.308537539      
P(z < z2) =    0.691462461      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.382924923   [ANSWER]

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