1. The mean and standard deviation of the grades of a group of students who took an economic exam were 69 and 7, respectively. The grades have a mound-shaped distribution. According to the Empirical Rule,
a) Approximately 68% of the grades were between which grade?
b) Approximately what percentage of grades were between 55 and 76?
a. 68% are mean ± 1 SD.
b. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores (from Z = -2 to +1).
To answer these questions, we can use the Empirical Rule, also known as the 68-95-99.7 rule, which applies to mound-shaped distributions that are approximately symmetric. This rule provides us with the percentages of data within certain standard deviations from the mean.
a) According to the Empirical Rule, approximately 68% of the grades fall within one standard deviation of the mean. Since the mean is 69 and the standard deviation is 7, we can calculate the range by adding and subtracting one standard deviation from the mean.
The lower limit would be:
69 - (1 * 7) = 62
And the upper limit would be:
69 + (1 * 7) = 76
Therefore, approximately 68% of the grades were between 62 and 76.
b) To find the percentage of grades between 55 and 76, we need to determine how many standard deviations each grade is away from the mean.
For 55:
Number of standard deviations away = (55 - 69) / 7 ≈ -2
For 76:
Number of standard deviations away = (76 - 69) / 7 ≈ 1
Now that we know the number of standard deviations, we can refer to the Empirical Rule:
Approximately 68% of the grades fall within one standard deviation of the mean. Therefore, 68% - 34% = 34% fall between one and two standard deviations from the mean.
Since our range covers approximately two standard deviations, we can estimate that:
Approximately 34% * 2 = 68% of the grades fall between 55 and 76.
Hence, approximately 68% of the grades were between 55 and 76.