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

Normally distributed means that median = mean = mode. Median = 50th percentile. Does that give you answer to a?

b. 50 + .5(68) = ?%
d. 50 - .5(68) = ?

Use similar procedures to answer the rest of the problems.

Normally distributed means that median = mean = mode. Median = 50th percentile. Does that give you answer to a?

To solve this problem, we will use the 68-95-99.7 rule, also known as the empirical rule or the three sigma rule. This rule states that in a normal distribution:

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

Given that the mean is 100 and the standard deviation is 20, we can calculate the following quantities:

a. Percentage of scores less than 100:
Since the mean is 100, this is equivalent to finding the percentage of scores within one standard deviation below the mean. According to the 68-95-99.7 rule, this percentage is approximately 34%.

b. Relative frequency of scores less than 120:
To find the relative frequency, we need to determine the number of standard deviations above the mean that corresponds to a score of 120. (120 - 100) / 20 = 1 standard deviation.
Since approximately 68% of scores fall within one standard deviation of the mean, the relative frequency of scores less than 120 is approximately 68%.

c. Percentage of scores less than 140:
To find the percentage of scores less than 140, we need to determine the number of standard deviations above the mean that corresponds to a score of 140. (140 - 100) / 20 = 2 standard deviations.
Since approximately 95% of scores fall within two standard deviations of the mean, the percentage of scores less than 140 is approximately 95%.

d. Percentage of scores less than 80:
To find the percentage of scores less than 80, we need to determine the number of standard deviations below the mean that corresponds to a score of 80. (80 - 100) / 20 = -1 standard deviation.
Since approximately 68% of scores fall within one standard deviation of the mean, the percentage of scores less than 80 is also approximately 68%.

e. Relative frequency of scores less than 60:
To find the relative frequency, we need to determine the number of standard deviations below the mean that corresponds to a score of 60. (60 - 100) / 20 = -2 standard deviations.
Since approximately 95% of scores fall within two standard deviations of the mean, the relative frequency of scores less than 60 is approximately 95%.

f. Percentage of scores greater than 120:
Since we are looking for scores greater than 120, we need to find the percentage of scores that fall more than one standard deviation above the mean. According to the 68-95-99.7 rule, this percentage is approximately 32%.
So, the percentage of scores greater than 120 is approximately 32%.

To answer these questions, first, let's understand 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 apply this knowledge to the given problem:

a. Percentage of scores less than 100:
Since the mean is 100, we know that the middle point of the distribution is 100. According to the 68-95-99.7 rule, about 50% of the data should be below the mean. Therefore, the percentage of scores less than 100 is approximately 50%.

b. Relative frequency of scores less than 120:
To find this, we need to consider the distance between the mean and the data point we are interested in, in terms of standard deviations. In this case, the distance is (120 - 100) / 20 = 1 standard deviation above the mean. According to the 68-95-99.7 rule, about 34% of the data falls between the mean and one standard deviation above the mean. Since we are only interested in the data below 120, we can estimate that the relative frequency of scores less than 120 is approximately 34%.

c. Percentage of scores less than 140:
Using the same approach as in part b, we find that the distance between 140 and the mean is (140 - 100) / 20 = 2 standard deviations above the mean. According to the 68-95-99.7 rule, we know that about 95% of the data falls within two standard deviations. Since we are interested in the data below 140, we can estimate that the percentage of scores less than 140 is approximately 95%.

d. Percentage of scores less than 80:
Similarly, the distance between 80 and the mean is (80 - 100) / 20 = -1 standard deviation below the mean. Using the 68-95-99.7 rule, about 34% of the data falls between the mean and one standard deviation below the mean. So, the percentage of scores less than 80 is approximately 34%.

e. Relative frequency of scores less than 60:
The distance between 60 and the mean is (60 - 100) / 20 = -2 standard deviations below the mean. Based on the 68-95-99.7 rule, we know that approximately 95% of the data falls within two standard deviations. Since we are interested in the data below 60, we can estimate that the relative frequency of scores less than 60 is approximately 95%.

f. Percentage of scores greater than 120:
Since we know that 68% of the data falls within one standard deviation of the mean, that leaves approximately 32% of the data outside of one standard deviation. Since we are interested in the data above 120, which is one standard deviation above the mean, we can estimate that the percentage of scores greater than 120 is approximately 32%.

Note that these values are estimations based on the 68-95-99.7 rule and may not be exact.