Specifications for a part for a DVD player state that the part should weigh betw

Question

Specifications for a part for a DVD player state that the part should weigh between 24.9 and 25.9 ounces. The process that produces the parts has a mean of 25.4 ounces and a standard deviation of .22 ounce. The distribution of output is normal. Use Table-A.

What percentage of parts will not meet the weight specs? (Round your "z" value and final answer to 2 decimal places. Omit the "%" sign in your response.)

Within what values will 99.74 percent of sample means of this process fall, if samples of n = 12 are taken and the process is in control (random)? (Round your answers to 2 decimal places.)

    
  Lower value?

, Upper value ?

What percentage of parts will not meet the weight specs? (Round your "z" value and final answer to 2 decimal places. Omit the "%" sign in your response.)

Explanation / Answer

A)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    24.9      
x2 = upper bound =    25.9      
u = mean =    25.4      
          
s = standard deviation =    0.22      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -2.27      
z2 = upper z score = (x2 - u) / s =    2.27      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.0116      
P(z < z2) =    0.9884      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.9768      

Thus, those outside this interval is the complement =    0.0232   = 2.32% [ANSWER]

**********************************

b)

The standard error in that case is

SE = s/sqrt(n) = 0.22/sqrt(12) = 0.06350853

As the middle area is          
          
Middle Area = P(x1<x<x2) =    0.9974      
          
Then the left tailed area of the left endpoint is          
          
P(x<x1) = (1-P(x1<x<x2))/2 =    0.0013      
          
Thus, the z score corresponding to the left endpoint, by table/technology, is          
          
z1 =    -3.01      
By symmetry,          
z2 =    3.01      
          
As          
          
u = mean =    25.4      
s = standard deviation =    0.06350853      
          
Then          
          
x1 = u + z1*s =    25.20883932      
x2 = u + z2*s =    25.59116068      

Hence, between 25.21 and 25.59. [ANSWER]

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