Assume womens ring sizes are normally distributed with a mea

Assume women’s ring sizes are normally distributed with a mean of 6 and a standard deviation of 1.0.

A. If one woman is randomly selected, find the probability that her ring size is < 6.2.

B. If 100 women are randomly selected, find the probability that the mean ring size is < 6.2.

C. A jeweler sees part b. and decides that all rings should be made < 6.2 because they would fit all but a few women with a ring guard. What’s wrong with this reasoning?

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    6.2      
u = mean =    6      
          
s = standard deviation =    1      
          
Thus,          
          
z = (x - u) / s =    0.2      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   0.2   ) =    0.579259709 [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 =    6.2      
u = mean =    6      
n = sample size =    100      
s = standard deviation =    1      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    2      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   2   ) =    0.977249868 [ANSWER]

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c)

Part b talks about the mean ring size of 100 women, not the individual women. Thus, the interpretation of the jeweler was wrong.