Question 1185493
the empirical rule states that:


68% of the scores are within 1 standard deviation from the mean.
95% of the scores are within 2 standard deviations from the mean.
99.7% of the scores are within 3 standard deviations from the mean.
50% of the scores are below the mean and 50% of the scores are above the mean.


from  this information, you can construct your own table as shown below:


34% of the scores are within 1 standard deviation below the mean and 34% of the scores are within 1 standard deviation above the mean.


47.5% of the scores are within 2 standard deviations below the mean and 47.5% of the scores are within 2 standard deviations above the mean.


49.85% of the scores are within 3 standard deviations below the mean and 49.85% of the scores are within 3 standard deviations above the mean.


from this information, you can solve this problem without the need of a calculator.


the mean is 74 and the standard deviation is 4.


50% of the scores are less than 74.


since 34% of the scores are within 1 standard deviation below the mean, and since 70 is 1 standard deviation below the mean, then 34% of the scores are between 70 and 74.
that means that 50 minus 34 = 16% are below 70.


since 47.5% of the scores are within 2 standard deviations above the mean, and since 82 - 74 = 8 = 2 standard deviations above the mean, then 47.5% of the scores are between 74 and 82.


add 50 to that and you get 97.5% of the scores are below 82.


your solutions are:


A) 50% of the exam scores are less than or equal to 74.
B) 16% of the exam scores are less than or equal to 70.
C) 97.5% of the exam scores are less than or equal to 82.


the following picture will show you what i mean.


<img src = "http://theo.x10hosting.com/2021/100301.jpg" >


from the diagram, the mean is at 0 which corresponds to the mean of74.


70 is 1 standard deviation below the mean.
66 is 2 standard deviations below the mean.
62 is 3 standard deviations below the mean.
below 62 is more than 3 standard deviations below the mean.


78 is 1 standard deviation above the mean.
82 is 2 standard deviations above the mean.
86 is 3 standard deviations above the mean
higher than 86 is more than 3 standard deviations above the mean.


50% of the scores are below 74.
that would be 34% + 13.5% + 2.35% + .15% counting from 75 down.
in a normal distribution, 50% of the scores are below the mean and 50% of the scores are above the mean.


16% of the scores are below 70.
that would be 13.5% + 2.35% + .15% counting from 70 down.


97.5% of the scores are below 82.
that would be 13.5% + 34% + 34% + 13.5% + 2.35% + .15% counting from 82 down.


to show you that these figures are reasonable, i used an online z-score calculator to give you the more detailed percentage, which you will find are close to the rough estimates.


<img src = "http://theo.x10hosting.com/2021/100302.jpg" >


<img src = "http://theo.x10hosting.com/2021/100303.jpg" >


<img src = "http://theo.x10hosting.com/2021/100304.jpg" >


multiply the answers in the calculator by 100 and they become percents.