Lengths of human pregnancies approximately follow a normal d

Lengths of human pregnancies approximately follow a normal distribution with a mean of 265 days and a standard deviation of 15 days.

__________ percent of human pregnancies are longer than 262 days.

Suppose that a random sample of 25 human pregnancies is considered. What is the probability that the sample will show an average length of between 259 and 273 days?

Solution

A)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    262      
u = mean =    265      
          
s = standard deviation =    15      
          
Thus,          
          
z = (x - u) / s =    -0.2      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.2   ) =    0.579259709 OR 57.9259709% [ANSWER]

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b)
  
We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    259      
x2 = upper bound =    273      
u = mean =    265      
n = sample size =    25      
s = standard deviation =    15      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -2      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    2.666666667      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.022750132      
P(z < z2) =    0.996169619      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.973419487   [ANSWER]