X is a normallydistributed random variable with a mean of 84

X is a normally-distributed random variable with a mean of 8.4 and standard deviation of 3.7. If b = 1.96 in the expression below, what should the value of a be? State your answer rounded to two decimal places.

X is a normally-distributed random variable with a mean of 7.8 and standard deviation of 2.8. If a = 12.5 in the expression below, what should the value of b be? State your answer rounded to two decimal places.

What is the probability that the standard normal variable z is between -1 and 2.2? State your answer rounded to four decimal places.

Solution

A)

As

x = u + z*sigma

Then

a = u + b*sigma

= 8.4 + 1.96*3.7

=15.652 [ANSWER]

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B)

As

z = (x - u) / sigma

Then

b = (a - u) / sigma

= (12.5 - 7.8)/2.8 = 1.678571429 [ANSWER]

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z1 = lower z score =    -1      
z2 = upper z score =     2.2      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.158655254      
P(z < z2) =    0.986096552      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.827441299 [ANSWER]