the scores on the Iowa basic skills reading text administere

the scores on the Iowa basic skills reading text administered to 5th graders are normally distributed with a mean if 5.5 and standard deviation of 0.5 (measured in reading level grade)

a) what is the probability one student at random will score 5.6 or higher?
b) a class of 20 students takes the exam and has a mean if 5.6. what is the probability a class of size 20 would have a mean if 5.6 or higher?
c) what conclusion would you make if a class did score that high?

Solution

A)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    5.6      
u = mean =    5.5      
          
s = standard deviation =    0.5      
          
Thus,          
          
z = (x - u) / s =    0.2      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.2   ) =    0.420740291 [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 =    5.6      
u = mean =    5.5      
n = sample size =    20      
s = standard deviation =    0.5      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.894427191      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.894427191   ) =    0.185546685

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c)

It\'s nothing special, it is not an unusual event, as 0.1855 is far from 0. This is not evidence that this class of 20 has significantly greater mean than all 5th grades in general.