A population of values has a normal distribution with =220.7=220.7 and =84.5=84.

Question

A population of values has a normal distribution with =220.7=220.7 and =84.5=84.5. You intend to draw a random sample of size n=68n=68.

Find P26, which is the score separating the bottom 26% scores from the top 74% scores.
P26 (for single values) =

Find P26, which is the mean separating the bottom 26% means from the top 74% means.
P26 (for sample means) =

Enter your answers as numbers accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Explanation / Answer

a)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.26      
          
Then, using table or technology,          
          
z =    -0.643345405      
          
As x = u + z * s,          
          
where          
          
u = mean =    220.7      
z = the critical z score =    -0.643345405      
s = standard deviation =    84.5      
          
Then          
          
x = critical value =    166.3373132   [ANSWER]

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b)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.26      
          
Then, using table or technology,          
          
z =    -0.643345405      
          
As x = u + z * s / sqrt(n)          
          
where          
          
u = mean =    220.7      
z = the critical z score =    -0.643345405      
s = standard deviation =    84.5      
n = sample size =    68      
Then          
          
x = critical value =    214.1075559   [ANSWER]  
  

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