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:

Suggest you make a drawing and label first…
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

f. Percentage of scores greater than 120

Calculate Z score of each value, then apply the rule.

Z = (score-mean)/SD

To answer these questions, we can use the 68-95-99.7 rule, also known as the empirical rule. This rule states that for a normal distribution:

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

Now, let's solve each question step by step:

a. Percentage of scores less than 100:
Since we are looking for scores less than 100, we are interested in the area to the left of 100 on the normal distribution curve. According to the empirical rule, 50% of the data lies to the left of the mean. Since the distribution is symmetric, 50% is evenly split between the left and right tails. Therefore, the percentage of scores less than 100 is 50%.

b. Relative frequency of scores less than 120:
To find the relative frequency of scores less than 120, we need to calculate the area to the left of 120 on the normal distribution curve. Since two standard deviations cover approximately 95% of the data, we can estimate that 95% / 2 = 47.5% of the data falls within one standard deviation of the mean in each direction. Therefore, the relative frequency of scores less than 120 is approximately 50% + 47.5% = 97.5%.

c. Percentage of scores less than 140:
Following the same logic, scores less than 140 would fall within two standard deviations of the mean. Therefore, we can estimate that the percentage of scores less than 140 is 50% + 47.5% + 47.5% = 145%.

d. Percentage of scores less than 80:
Similarly, scores less than 80 would fall within one standard deviation to the left of the mean. So, we can estimate that the percentage of scores less than 80 is 50% - 47.5% = 2.5%.

e. Relative frequency of scores less than 60:
We need to move three standard deviations to the left of the mean to find the area less than 60 on the normal distribution curve. Since 99.7% of the data falls within three standard deviations, the relative frequency of scores less than 60 is approximately 50% - 47.5% - 47.5% - 2.5% = 2.5%.

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 from 100%. Using the same logic as in question b, the percentage of scores greater than 120 is 100% - 97.5% = 2.5%.