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Empirical Rule: Mean is 84 with a standard deviation of 7. Approximately how much is between 63 and 98?

I would like to know how to go about getting the answer to these questions.
 
 
1. Empirical Rule: Mean is 84 with a standard deviation of 7. Approximately how much is between 63 and 98?
 
 
2. Empirical Rule: Mean is 27 with a standard deviation of 3. Approximately how much of the distribution is between 24 and 36?
 
 
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3 Answers

For a standard bell-shape curve, the distribution:
a) From the Mean to 1 sdev = 36.1%
b) between 1sdev and 2sdev = 13.6%
c) between 2sdev and 3sdev = 2.1%
D) between 3sdev and 4sdev = 0.1%
 
1) Given the Mean is 84 and the sdev=7
98 is 2 sdev away from 84 and the distribution is:
34.1% (between the mean and 1sdev) + 13.6% (between 1sdev and 2sdev)
 
63 is 3 sdev away from 84 and the distribution is:
34.1% + 13.6% + 2.1%
 
Between 63 and 98, the total distribution is 97.5%
 
2) Given the Mean is 27 and the sdev = 3
36 is 3 sdev away from 27 and its distribution is  36.1% + 13.6% + 2.1%
24 is 1 sdev away from 27 and its distribution is 36.1%
 
Between 24 and 36, the total distribution is 87.9%
Use the standartdised normal distribution with the variable z = (x-x0)/σ, where x - current variable (63 and 98), and x0= 84, σ =7 (in this case mean = 0 and the new standard deviation = 1), and calculate the probability using standard tables for integral of errors.
Say you survey a bunch of people and get their numerical opinions on 1 thing.  You then do some statistics and you find the mean and standard deviation.
 
The empirical rule says that 99.7% of your results will within 3 standard deviations above or 3 standard deviations below your mean.  
 
Say your mean is 20 and standard deviation is 2. Then 99.7% of your results will fall between 
20 - 2 - 2 - 2 = 14 and 20 +2+2+2 = 26.  So between 14 and 26. 
 
The empirical rule also says that 95% of your results will be with 2 standard deviation above or 2 standard deviations below your mean. 
And the Emp. rule says that 68% of the results will be within 1 standard deviation of your mean.
 
Since this is a "normal distribution" the % results will also be symmetrical.  Meaning half of the 68% will be below the mean and half of the 68% will be above the mean. The same goes for the 95% and the 99.7%. 
 
So let's get to the problems:
1. if your mean is 84 and standard deviation is 7 then:
84 - 7 = 77 and 84 + 7 = 91 68% of your results will be between 77 and 91.  34% above up to 91 and 34% below down to 77.
 
84-7-7 = 70 and 84+7+7 = 98.  95% of your results will be between 70 and 98.  47.5% above and 47.5% below.
 
84-7-7-7 = 63 and 84+7+7+7 = 105.  99.7% of your results will be between 63 and 105. 49.85% above and 48.85% below.
 
Remember how I said this is symmetrical. 
Since 98 is the upper half of 95% that would be 95/2 = 47.5%
63 is the lower half of 99.7% that would be 99.7/2 = 49.85%
Add them together to get teh upper and the lower of your mean and you get 47.5+49.85 = 97.35%
 
2) 27 - 3 = 24, so 24 is 1 standard deviation below.( 68%/2 because of lower half)
27+3+3+3 = 36, so 36 is 3 standard deviations above.( 99.7%/2 because of upper half)
 
34% + 49.85% = 83.85%

Comments

If the question wants you to use the Empirical Rule, then you should be using the 99.7%, 95%, 68% split.  Yes you can use the formulas and chart and find a more exact answer but the question is not asking for that.

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