Use the normal distribution of SAT critical reading scores f

Use the normal distribution of SAT critical reading scores for which the mean is 509 and the standard deviation is 114. Assume the variable x is normally distributed. What percent of the SAT verbal scores are less than 675? If 1000 SAT verbal scores are randomly selected, about how many would you expect to be greater than 575?

Solution

A)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    675      
u = mean =    509      
          
s = standard deviation =    114      
          
Thus,          
          
z = (x - u) / s =    1.456140351      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   1.456140351   ) =    0.927323088 = 92.73% [answer]

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B)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    575      
u = mean =    509      
          
s = standard deviation =    114      
          
Thus,          
          
z = (x - u) / s =    0.578947368      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.578947368   ) =    0.281312343

So we expect 0.281312343*1000 = 281.312343 = 281 scores [ANSWER, 281]