The foreman of a bottling plant has observed that the amount

The foreman of a bottling plant has observed that the amount of soda in each 32-ounce bottle is actually a normally distributed random variable, with a mean of 32.2 ounces and a standard deviation of 0.3 ounce.

If a customer buys one bottle, what is the probability that the bottle will contain more than 32 ounces?

If a customer buys a cartoon of four bottles. What is the probability that the mean amount of the four bottles will be greater than 32 ounces?

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    32      
u = mean =    32.2      
          
s = standard deviation =    0.3      
          
Thus,          
          
z = (x - u) / s =    -0.666666667      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.666666667   ) =    0.747507462 [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 =    32      
u = mean =    32.2      
n = sample size =    4      
s = standard deviation =    0.3      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -1.333333333      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -1.333333333   ) =    0.90878878 [ANSWER]