SOLUTION: The reading test scores for the population of fifth graders is normally distributed with a mean of of 55 and a standard deviation of 8. What percentage of fifth graders score more

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Question 1198914: The reading test scores for the population of fifth graders is normally distributed with a mean of of 55 and a standard deviation of 8. What percentage of fifth graders score more than 47, but less than 71?
The correct answer is one of the following. Which one is correct?
A) 13.5%
B) 84%
C) 97.5%
D) 81.5%

Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
mean = 55
standard deviation = 8
want probability of scores between 47 and 71.

probability is .8166

use z-score if you don't have a calculator that does it directly.

z-score formula is z = (x - m) / s
x is the raw score
m is the mean
s is the standard deviation.

on the low side, formula becomes z = (47 - 55)/8 = -1
on the high side, formula becomes z = (71 - 55)/8 = 2

use a calculator or a z-score table to find the area to the left of those z-scores.

area to the left of z-score of -1 = .1586552
agea to the left of z-score of 2 = .9772499
area in between = larger area minus smaller area = .81859467
round to 4 decimal places to get .8186

the probability of getting a score between 47 and 71 is equal to .8186.
that's your answer.

a calculator i used to get the answer directly is found at https://davidmlane.com/hyperstat/z_table.html

here are the results from using that calculator.



calculator i used was the ti-84 plus.

you could also use the z-score table, but a calculator is much easier.


Answer by ikleyn(52817)   (Show Source): You can put this solution on YOUR website!
.

Regarding this problem, I want to make some notices. 

    There is a dark spot in the problem. 

    This dark spot is that the problem does not say if we should consider scores 
    as continues variables or as discrete integer variables.


    If the scores are integer variables, then the interval "more than 47, but less than 71" 

    is the interval [48,70], but not the interval [47,71], which @Theo considers in his solution.


    Also, if the scores are discrete integer variables, then we should use the "continuous correction",

    which leads to the interval [47.5,70.5].

Since there is this dark spot in the problem's formulation, I even do not try to give a solution.

The formulation must be clarified, otherwise any solution is questionable.



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