Assume that the sick leave, X, taken by the typical worker per year has mean µ =

Question

Assume that the sick leave, X, taken by the typical worker per year has mean µ = 10 and standard deviation = 2 measured in days. A firm has n = 20 employees. (A) Assuming independence, use the Central limit theorem to find the approximate probability that mean number X of sick days per year of the 20 employees from the firm exceeds 11? (B) Assuming independence, how many sick days should the firm budget if the financial of- ficer wants the probability of exceeding the budgeted days to be less than 0.2?

Explanation / Answer

a)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    11      
u = mean =    10      
n = sample size =    20      
s = standard deviation =    2      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    2.236067977      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.236067977   ) =    0.012673659 [ANSWER]

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b)

First, we get the z score from the given left tailed area. As          
          
Left tailed area = 1 -0.2 =   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 =    2      
n = sample size =    20      
Then          
          
x = critical value =    10.37638446   [ANSWER]  

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