A binomial probability distribution has p 20 and n 100 If

A binomial probability distribution has p = .20 and n = 100. If required, round your answers to four decimal places. What are the mean and standard deviation? Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? What is the probability of exactly 24 successes? What is the probability of 17 to 23 successes? What is the probability of 14 or fewer successes?

Solution

a)

mean = np = 100*0.20 = 20

standard deviation = sqrt(np(1-p)) = sqrt(100*0.20*(1-0.20)) = 4 [ANSWER]

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b)

Yes, becasue np > 10, and nq = 80 >10 as well.

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c)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    23.5      
x2 = upper bound =    24.5      
u = mean =    20      
          
s = standard deviation =    4      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    0.875      
z2 = upper z score = (x2 - u) / s =    1.125      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.809213047      
P(z < z2) =    0.869705483      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.060492436   [ANSWER]

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d)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    16.5      
x2 = upper bound =    23.5      
u = mean =    20      
          
s = standard deviation =    4      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.875      
z2 = upper z score = (x2 - u) / s =    0.875      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.190786953      
P(z < z2) =    0.809213047      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.618426094   [ANSWER]

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e)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    14.5      
u = mean =    20      
          
s = standard deviation =    4      
          
Thus,          
          
z = (x - u) / s =    -1.375      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -1.375   ) =    0.084565722 [ANSWER]