A sample of 3000 observations has a mean of 86 and a standard deviation of 13. The distribution is bell-shaped. Using the empirical rule, find what percentage of the observations fall in the intervals
What intervals?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities between your Z scores.
To find the percentage of observations that fall within certain intervals using the empirical rule, we need to consider the standard deviations from the mean. The empirical rule states that for a bell-shaped distribution, approximately 68% of the observations fall within one standard deviation of the mean, approximately 95% fall within two standard deviations, and approximately 99.7% fall within three standard deviations.
Given that the sample has a mean of 86 and a standard deviation of 13, we can apply these percentages to calculate the intervals.
1. Within one standard deviation:
To find the interval within one standard deviation, we can calculate the range by adding and subtracting one standard deviation from the mean.
Lower limit = mean - 1 * standard deviation = 86 - 1 * 13 = 73
Upper limit = mean + 1 * standard deviation = 86 + 1 * 13 = 99
So, approximately 68% of the observations fall within the interval [73, 99].
2. Within two standard deviations:
To find the interval within two standard deviations, we can calculate the range by adding and subtracting two standard deviations from the mean.
Lower limit = mean - 2 * standard deviation = 86 - 2 * 13 = 60
Upper limit = mean + 2 * standard deviation = 86 + 2 * 13 = 112
So, approximately 95% of the observations fall within the interval [60, 112].
3. Within three standard deviations:
To find the interval within three standard deviations, we can calculate the range by adding and subtracting three standard deviations from the mean.
Lower limit = mean - 3 * standard deviation = 86 - 3 * 13 = 47
Upper limit = mean + 3 * standard deviation = 86 + 3 * 13 = 125
So, approximately 99.7% of the observations fall within the interval [47, 125].
These intervals provide an estimate of the percentage of observations that fall within certain ranges based on the empirical rule for a bell-shaped distribution.