Question 1147159: Assume that a set of test scores is normally distributed with a mean of
80
80 and a standard deviation of
25
25. Use the 68-95-99.7 rule to find the following quantities.
1. The percentage of scores less than
80?
2.The percentage of scores greater than
105
3.The percentage of scores between
30 and 105
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the 68-95-99.7 rule says:
68% of the area under the normal distribution curve is within plus or minus 1 standard deviation from the mean.
this means that 34% is within 1 standard deviation above the mean and 34% is within 1 standard deviation below the mean.
95% of the area under the normal distribution curve is within plus or minus 2 standard deviations from the mean.
this means that 49.5% is within 2 standard deviations above the mean and 49.5% is within 2 standard deviations below the mean.
99.7% of the area under the normal distribution curve is within plus or minus 3 standard deviation from the mean.
this means that 49.85% is within 3 standard deviations above the mean and 49.85% is within 3 standard deviations below the mean.
since we are dealing with a normal disribution, we know that 50% of the area under the normal distribution curve is above the mean and 50% is below.
this tells us that.
34% of the area under the normal distribution curve is within 1 standard deviation above the mean, leaving 16% of the area under the normal distribution curve beyond 1 standard deviation above the mean.
49.5% of the area under the normal distribution curve is within 2 standard deviations above the mean, leaving .5% of the area under the normal distribution curve beyond 2 standard deviations above the mean.
49.85% of the area under the normal distribution curve is within 3 standard deviations above the mean, leaving .15% of the area unde rthe normal distribution curve beyond 3 standard deviations above the mean.
since the normal distribution curve is symmetric about the mean, the same applies to standard deviations below the mean.
we can use this information to answer the questions.
question 1:
what is the percentage of scores below 80?
since the area below the mean will always be 50%, this is answered simply by the basic properties of the normal distribution which states that the mean is exactly in the middle of the normal distribution curve.
you could state that 68% of the area is within 1 standard deviation, leaving 34% not within 1 standard deviation, for a total of 100%.
divide that by two and you get 50% above and 50% below.
the answer, however, relies on the fact that we know that the normal distribution curve is symmetric about the mean and we know that the mean is in the middle and that the area above the mean is the same as the area below he mean.
this means that plus or minus 68% means that 34% is above and 34% is below, with similar distributions for 2 and 3 standard deviations.
question 2:
what is the percentage of scores greater than 105?
z-score formula helps here.
z = 105 - 80) / 25 = plus 1
z of plus 1 means 1 standard deviation above the mean.
since plus or - 1 = 68%, then plus 1 = 34% beetween the men and 1 standad deviation above the mean.
the area above that would be 50% - 34% = 16%.
this is your solution.
question 3:
what is the percentage of scores between 30 and 105?
z-score help hear.
z-score for 105 is (105 - 80) / 25 = 1
z-score for 30 is (30 - 80) / 25 = -2
this says that 105 is 1 standard deviation above the mean and 30 is 2 standard deviations below the mean.
you want the area in between.
30 is 2 standard deviations below the mean = 49.5% of the area under the normal distribution curve.
105 is 1 standard deviation above the mean = 34% of the area under the normal distribution curve.
add these areas up and you get a total of 83%% of the area under the normal distribution curve.
a look at the following displays will show this to be a resonable assumption.
the first display shows 80% below 80.
the second display shows 15.87% above 105.
this is very close to 16% that we calculated from the given information.
the third display shows 81.86% between 30 and 105.
this is, again, very close to 83.5% that we calculated from the given information.
these will not be right on to our manually calculated numbers because our manually calculated numbers are not right on to the actual percentages.
the actual percentages are:
within 1 standard deviation equals 68.27%
within 2 standard deviations equals 95.45%
within 3 standard deviations equals 99.73%
the displays use the actuals rather than the percentages from the rule.
the rule, however, is pretty accurate and very useful, given that we might not have access to, or know, the actual percentages.
|
|
|
