The 68-95-99.7 Rule

About 68% of the data points fall within 1 standard deviation of the mean

About 95% of the data points fall within 2 standard deviation of the mean

About 99.7% of the data points fall within 3 standard deviations of the mean

Assume that a set test scores is normally distributed with a mean of 100 and a standard deviation of 20. Use the 68-95-99.7 rule to find the following

1) Percentages of scores less than 100
2) Percentage of scores less than 140
3) Percentage of scores less than 80
4) Percentage of scores between 80 and 120
5) Percentage of scores between 80 and 140

Assume the resting heart rates for a sample of individuals are normally distributed with a mean of 70 and a standard deviation of 15. Use the 68-95-99.7 rule to find the following
1) Percentage of rates less than 70
2) Percentage of rates less than 55
3) Percentage of rates less than 85
4) Percentage of rates greater than 85
5) Percentage of rates greater than 55
6) Percentage of rates between 55 and 100
7) Percentage of rates between 70 and 100

1) Percentages of scores less than 100: 50%

2) Percentage of scores less than 140: 97.5%

3) Percentage of scores less than 80: 16%

4) Percentage of scores between 80 and 120: 68%

5) Percentage of scores between 80 and 140: 81.5%

1) Percentage of rates less than 70 : 50%

2) Percentage of rates less than 55 : 16%

3) Percentage of rates less than 85 : 97.5%

4) Percentage of rates greater than 85 : 2.5%

5) Percentage of rates greater than 55 : 84%

6) Percentage of rates between 55 and 100: 81.5%  

7) Percentage of rates between 70 and 100: 47.5%

Step-by-step explanation:

We have a random variable normally distributed with a mean of 100 and a standard deviation of 20.

1) Percentages of scores less than 100: 50%

As the mean is 100, 50% of the data lies below 100.

2) Percentage of scores less than 140: 97.5%

The data is what lies below (mean +2 sd). In this case, applies the 95% rule for the higher scores (above 100), which means we have 95/2=47.5 of the data between 100 and 140.

The data below 100 represents 50%.

So the scores under 140 are 50+47.5=97.5%.

3) Percentage of scores less than 80: 16%

The scores under 80 are a (mean-1 sd).

This means that is half of the data, less 68/2=34 (the area that is under the first standard deviation of the mean and the mean).

Then, the scores under 80 are 50-34=16%.

4) Percentage of scores between 80 and 120: 68%

The scores are under one deviation of the mean (to both sides). The 68% rule applies.

5) Percentage of scores between 80 and 140: 81.5%

The lower scores, between 80 and 100 are in the area between one deviation and the mean, so it has a percentage of 68/2=34%.

The higher scores are 2 deviations frome the mean, so they have 95/2=47.5% of the scores.

Between 80 and 140 are 34+47.5=81.5% of the scores.

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step-by-step explanation:

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