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 100b.Relative frequency of scores less than 120c.Percentage of scores less than 140d.Percentage of scores less than 80e.Relative frequency of scores less than 60f.Percentage of scores greater than 1

a. In a normal distribution, mean = median = mode. Hint: What is median?

b. median + 1/2(68)

Use similar process for other questions.

f. all of the scores > 1. (> = greater than)

The mean is 100 with standard deviation of 15.the results come very close to fitting a normal curve

Find the percent of students whose score wil fall in less than 100

To answer these questions using the 68-95-99.7 rule, we'll need to understand the concept of standard deviations and how they relate to the normal distribution.

First, let's draw a normal distribution curve with a mean of 100 and a standard deviation of 20. The curve should be symmetrical, with the highest point (peak) at the mean and the tails extending indefinitely.

a. Percentage of scores less than 100: Since the mean is 100, the percentage of scores less than 100 is 50%. This is because the normal distribution is symmetrical and the mean is the midpoint of the distribution.

b. Relative frequency of scores less than 120: According to the 68-95-99.7 rule, approximately 68% of the scores fall within one standard deviation of the mean. So, within the interval of 100 - 1(20) = 80 to 100 + 1(20) = 120, approximately 68% of the scores fall in this range.

c. Percentage of scores less than 140: The interval from the mean of 100 to 140 is two standard deviations above the mean. According to the 68-95-99.7 rule, approximately 95% of the scores fall within two standard deviations of the mean. Therefore, approximately 95% of the scores are less than 140.

d. Percentage of scores less than 80: Again, using the rule, since the interval from the mean of 100 to 80 is one standard deviation below the mean, approximately 68% of the scores are less than 80.

e. Relative frequency of scores less than 60: For this, we need to calculate how many standard deviations below the mean 60 is. The difference between 60 and 100 is 100 - 60 = 40. Since the standard deviation is 20, 40/20 = 2 standard deviations below the mean. According to the 68-95-99.7 rule, approximately 95% of the scores fall within two standard deviations of the mean. Therefore, approximately 2.5% of the scores are less than 60.

f. Percentage of scores greater than 1: Since 1 is significantly below the standard deviation and mean, it is safe to assume that nearly 100% of scores will be greater than 1.

Remember, these calculations are approximate and are based on the 68-95-99.7 rule for normally distributed data.