Assume that a set of test scores is normally distributed with a mean of 8080 and a standard deviation of 55. Use the 68-95-99.7 rule to find the following quantities.
Use the rule they suggest. It involves 1,2,3 sigma intervals
What following quantities?
To use the 68-95-99.7 rule, also known as the empirical rule, we can determine the percentage of values that fall within a certain range of standard deviations from the mean in a normal distribution.
1. Percentage of values within one standard deviation of the mean:
According to the 68-95-99.7 rule, approximately 68% of the values in a normal distribution lie within one standard deviation of the mean. So, in this case, we can calculate the range by adding and subtracting one standard deviation (55) from the mean (8080):
Mean ± 1 standard deviation = 8080 ± 55
Therefore, the range would be (8025, 8135).
2. Percentage of values within two standard deviations of the mean:
According to the 68-95-99.7 rule, approximately 95% of the values in a normal distribution lie within two standard deviations of the mean. So, we can calculate the range by adding and subtracting two standard deviations (55 * 2) from the mean:
Mean ± 2 standard deviations = 8080 ± 110
Therefore, the range would be (7970, 8190).
3. Percentage of values within three standard deviations of the mean:
According to the 68-95-99.7 rule, approximately 99.7% of the values in a normal distribution lie within three standard deviations of the mean. So, we can calculate the range by adding and subtracting three standard deviations (55 * 3) from the mean:
Mean ± 3 standard deviations = 8080 ± 165
Therefore, the range would be (7915, 8245).
Please note that these percentages are approximations based on the empirical rule and may not be exact in every scenario.