College

Use the Empirical Rule to answer the following questions about the test scores: 56, 63, 70, 77, 84, 91, 98.

Given:
- The mean is 77.
- The standard deviation is 7.

Questions:
a. What percentage of the test scores are between 70 and 84?
b. What percentage of the test scores are between 63 and 91?
c. What percentage of the test scores are between 56 and 98?
d. What percentage of the test scores are between 77 and 84?
e. What percentage of the test scores are less than 77?
f. What percentage of the test scores are less than 84?

Answer :

Sure! Let's use the Empirical Rule, which applies to data that follows a normal distribution. This rule is also known as the 68-95-99.7 rule because it states that:

1. About 68% of the data falls within 1 standard deviation from the mean.
2. About 95% of the data falls within 2 standard deviations from the mean.
3. About 99.7% of the data falls within 3 standard deviations from the mean.

Let's apply the Empirical Rule to the problem with the given mean and standard deviation:

a. Given: The mean is 77, and the standard deviation is 7.

b. Calculations:

c. Test scores between 70 and 84:
- 70 is one standard deviation below the mean (77 - 7 = 70).
- 84 is one standard deviation above the mean (77 + 7 = 84).
- According to the Empirical Rule, approximately 68% of the scores are between 70 and 84.

d. Test scores between 63 and 91:
- 63 is two standard deviations below the mean (77 - 27 = 63).
- 91 is two standard deviations above the mean (77 + 2
7 = 91).
- Approximately 95% of the scores are between 63 and 91.

e. Test scores between 56 and 98:
- 56 is three standard deviations below the mean (77 - 37 = 56).
- 98 is three standard deviations above the mean (77 + 3
7 = 98).
- Approximately 99.7% of the scores are between 56 and 98.

f. Test scores between 77 and 84:
- 77 is the mean.
- 84 is one standard deviation above the mean.
- For the range from the mean to one standard deviation above the mean, about 34% of the scores fall between 77 and 84 (half of the 68%).

g. Test scores less than 77:
- 77 is the mean.
- Since the mean divides the normal distribution in half, 50% of the scores are less than 77.

h. Test scores less than 84:
- 84 is one standard deviation above the mean.
- The percentage of scores less than 84 includes all scores below the mean and half of the scores between the mean and one standard deviation above the mean.
- Therefore, 50% (below the mean) plus 34% (from 77 to 84) gives us 84%.

By understanding and applying the Empirical Rule, we can accurately describe the distribution of these test scores.