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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 + 27 = 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 + 37 = 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.
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 + 27 = 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 + 37 = 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.
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