The number of pieces of mail that a department receives each

The number of pieces of mail that a department receives each day can be modeled by a distribution having mean 44 and variance 64. For a random sample of 36 days, what can be said about the probability that the sample mean will be less than 40 or more then 48 using the central limit theorem Chebyshev\'s theorem

Solution

6.

a)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    40      
x2 = upper bound =    48      
u = mean =    44      
n = sample size =    36      
s = standard deviation =    8      
          
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 =    3      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.001349898      
P(z < z2) =    0.998650102      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.997300204      

Thus, those outside this interval is the complement =    0.002699796   [ANSWER]

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b)

As we can see, the z scores are -3 and 3 here.

By Chebyshev\'s theorem, those more than k = 3 standard deviations away from the mean is at most 1/k^2, or

1/k^2 = 1/3^2 = 0.11111111 [ANSWER]