Answer :
These are the solution of the given condition:
1. For a class average of 68% and a standard deviation of 5%, around 4.7% of grades are above 83, below 53, and beyond 3 standard deviations.
2. About 68% of grades fall between 63 and 73, 95% between 63 and 78, and 99.7% between 63 and 83, based on the rule.
3. Roughly 16.3% of grades are below 58 or above 83, estimated by combining percentages beyond 2 standard deviations and beyond 3 standard deviations.
The Empirical Rule, also known as the 68-95-99.7 rule, is a guideline for the approximate percentage of data that falls within certain standard deviation intervals of a normal distribution. According to this rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% falls within two standard deviations.
- Approximately 99.7% falls within three standard deviations.
Given that the class average is 68% and the standard deviation is 5%, we can use the Empirical Rule to answer the questions:
a. What percentage of the grades are above 83?
Since 83 is 3 standard deviations above the mean (68 + 3 * 5), we can estimate that about 99.7% - 95% = 4.7% of the grades are above 83.
b. What percentage of the grades are below 53?
Since 53 is 3 standard deviations below the mean (68 - 3 * 5), we can estimate that about 99.7% - 95% = 4.7% of the grades are below 53.
c. What percentage of the grades are between 63 and 73?
Between 63 and 73 is within one standard deviation of the mean, so approximately 68% of the grades are between 63 and 73.
d. What percentage of the grades are between 63 and 78?
Between 63 and 78 is within two standard deviations of the mean, so approximately 95% of the grades are between 63 and 78.
e. What percentage of the grades are between 63 and 83?
Between 63 and 83 is within three standard deviations of the mean, so approximately 99.7% of the grades are between 63 and 83.
f. What percentage of the grades are less than 58 or more than 83?
This can be calculated as the sum of the percentages less than 58 and the percentage more than 83.
- Less than 58 is approximately 50% - 34% = 16% (since 58 is 2 standard deviations below the mean).
- More than 83 is approximately 100% - 99.7% = 0.3% (since 83 is 3 standard deviations above the mean).
Adding these percentages, we get approximately 16% + 0.3% = 16.3%.
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