Research suggests Americans make an average of 10 phone call

Research suggests Americans make an average of 10 phone calls per day. Let the number of calls be normally distributed with a standard deviation of 3 calls.

What is the probability that an average American makes between 8 and 14 calls per day? What is probability make less than 7 calls? 80% make at most how many calls?

Solution

a)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    8      
x2 = upper bound =    14      
u = mean =    10      
          
s = standard deviation =    3      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.666666667      
z2 = upper z score = (x2 - u) / s =    1.333333333      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.252492538      
P(z < z2) =    0.90878878      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.656296243   [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    7      
u = mean =    10      
          
s = standard deviation =    3      
          
Thus,          
          
z = (x - u) / s =    -1      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z >   -1   ) =    0.158655254 [answer]

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c)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.8      
          
Then, using table or technology,          
          
z =    0.841621234      
          
As x = u + z * s / sqrt(n)          
          
where          
          
u = mean =    10      
z = the critical z score =    0.841621234      
s = standard deviation =    3      
          
Then          
          
x = critical value =    12.5248637   [ANSWER]