The American College Testing Program designs and produces th

The American College Testing Program designs and produces the ACT test, a college entrance exam. The scores on the ACT exam for incoming freshmen at the University of Iowa are approximately normally distributed with mean 23.2 and standard deviation 3.1.

a) What is the probability that a randomly selected freshman has an ACT score exceeding 26?

b) If five freshmen are randomly selected, what is the chance that all five will have ACT scores exceeding 26? (Hint: For each freshman, there are only two possibilities – his or her score either exceeds 26 or it is 26 or lower.)

c) If five freshmen are randomly selected, what is the chance that exactly one has a score exceeding 26?

Please show work

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    26      
u = mean =    23.2      
          
s = standard deviation =    3.1      
          
Thus,          
          
z = (x - u) / s =    0.903225806      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.903225806   ) =    0.183203032 [ANSWER]

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b)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    5      
p = the probability of a success =    0.183203032      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.000206378 [ANSWER]

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c)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    5      
p = the probability of a success =    0.183203032      
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.40771729 [ANSWER]