Z is a standard normal random variable. Compute the following probabilities. a.

Question

Z is a standard normal random variable. Compute the following probabilities. a. P(-1.33 lessthanorequalto Z lessthanorequalto 1.67) b. P(1.23 lessthanorequalto Z lessthanorequalto 1.55) c. P(Z Greaterthanorequalto 2.32) d. P(Z Greaterthanorequalto -2.08) e. P(Z lessthanorequalto -1.08) 2. A professor at a local community college noted that the grades of his students were normally distributed with a mean of 74 and a standard deviation of 10. The professor has informed us that 6.3 percent of his students received A's while only 2.5 percent of his students failed the course and received F's. a. What is the minimum score needed to make an A? b. What is the maximum score among those who received an F? c. If there were 5 students who did not pass the course, how many students took the course?

Explanation / Answer

1.

A)

z1 = lower z score =    -1.33      
z2 = upper z score =     1.67      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.091759136      
P(z < z2) =    0.952540318      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.860781183   [ANSWER]

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b)

z1 = lower z score =    1.23      
z2 = upper z score =     1.55      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.890651448      
P(z < z2) =    0.939429242      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.048777794   [ANSWER]

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c)

Using a table/technology, the left tailed area of this is          
          
P(z <   2.32   ) =    0.989829561
          
Thus, the right tailed area is the complement,          
P(z >   2.32   ) =    0.010170439 [ANSWER]

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d)

Using a table/technology, the left tailed area of this is          
          
P(z <   -2.08   ) =    0.018762766
          
Thus, the right tailed area is the complement,          
P(z >   -2.08   ) =    0.981237234 [ANSWER]

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e)

Using a table/technology, the left tailed area of this is          
          
P(z <   -1.08   ) =    0.14007109 [ANSWER]

     

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