Skull size of a population of rats follows a normal populati

Skull size of a population of rats follows a normal population with std dev = 10mm. Y bar = mean skull size of a random sample of 64 individuals from the population. µ = population mean skull size. Find the two probabilities

P(µ - 2.5mm < Ybar < µ + 2.5mm)

P(µ - 3.75mm < Ybar < µ + 1.25mm)

Solution

a)

Note that the mean of Ybar - u = 0.

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    -2.5      
x2 = upper bound =    2.5      
u = mean =    0      
n = sample size =    64      
s = standard deviation =    10      
          
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      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.022750132      
P(z < z2) =    0.977249868      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.954499736   [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 =    -3.75      
x2 = upper bound =    1.25      
u = mean =    0      
n = sample size =    64      
s = standard deviation =    10      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -3      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    1      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.001349898      
P(z < z2) =    0.841344746      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.839994848   [ANSWER]