Q The overhead reach distances of adult females are normally

Q. The overhead reach distances of adult females are normally distributed with a mean of 195cm and a standard deviation of 8.9 cm.

a. Find the probability that an individual distance is greater than 204.30cm.

b. Find the probability that the mean for 20 randomly selected distances is greater than 192.80 cm.

c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?

     A. The normal distribution can be used because the finite population correction factor is small.

     B. The normal distribution can be used because the probability is less than 0.5

     C. The normal distribution can be used because the original population has a normal distribution.

     D. The normal distribution can be used because the mean is large.

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    204.3      
u = mean =    195      
          
s = standard deviation =    8.9      
          
Thus,          
          
z = (x - u) / s =    1.04494382      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.04494382   ) =    0.148024468 [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 =    192.8      
u = mean =    195      
n = sample size =    20      
s = standard deviation =    8.9      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -1.105471809      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -1.105471809   ) =    0.865522398 [ANSWER]

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c)

C. The normal distribution can be used because the original population has a normal distribution.