Using the 68-95-99.7 rule:

Assume that a set of test scores is normally distributed with a mean of 100 and a standard deviation of 20. Use the 68-95-99.7 rule to find the following quantities:
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a. Percentage of scores less than 100 =
b. Relative frequency of scores less than 120 =
c. Percentage of scores less than 140 =
d. Percentage of scores less than 80 =
e. Relative frequency of scores less than 60 = 32
f. Percentage of scores greater than 120 =

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To answer these questions using the 68-95-99.7 rule, we need to understand that the rule states that for a normal distribution:

- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% falls within two standard deviations of the mean.
- Approximately 99.7% falls within three standard deviations of the mean.

Let's find the quantities:

a. Percentage of scores less than 100:
Since the mean is 100, any score less than 100 falls within one standard deviation below the mean. According to the 68-95-99.7 rule, approximately 68% of the scores fall within one standard deviation of the mean. Therefore, the percentage of scores less than 100 is approximately 68%.

b. Relative frequency of scores less than 120:
To find the relative frequency, we need to determine how many standard deviations away 120 is from the mean and then use the rule. The mean is 100 and the standard deviation is 20. So, (120 - 100) / 20 = 1 standard deviation. According to the 68-95-99.7 rule, approximately 68% of the data falls within one standard deviation of the mean. Therefore, the relative frequency of scores less than 120 is approximately 68%.

c. Percentage of scores less than 140:
Using the same logic as above, (140 - 100) / 20 = 2 standard deviations away from the mean. According to the 68-95-99.7 rule, approximately 95% of the data falls within two standard deviations of the mean. Therefore, the percentage of scores less than 140 is approximately 95%.

d. Percentage of scores less than 80:
Using the same logic as above, (80 - 100) / 20 = -1 standard deviation away from the mean. Since we're looking for scores less than 80, we need to consider the area to the left of -1 standard deviation. According to the 68-95-99.7 rule, the area to the left of -1 standard deviation is the same as the area to the right of 1 standard deviation. So, approximately 68% of the data falls within one standard deviation of the mean. Therefore, the percentage of scores less than 80 is approximately 68%.

e. Relative frequency of scores less than 60:
To find the relative frequency, we need to determine how many standard deviations away 60 is from the mean. (60 - 100) / 20 = -2 standard deviations away from the mean. Since we're looking for scores less than 60, we need to consider the area to the left of -2 standard deviations. According to the 68-95-99.7 rule, the area to the left of -2 standard deviations is the same as the area to the right of 2 standard deviations, which is approximately 95%. Therefore, the relative frequency of scores less than 60 is 95%.

f. Percentage of scores greater than 120:
To find the percentage of scores greater than 120, we can subtract the percentage of scores less than 120 (calculated in part b) from 100%. Since the relative frequency of scores less than 120 is approximately 68%, the percentage of scores greater than 120 would be 100% - 68% = 32%.