Assume that a set of test scores is normally distributed with a mean of 100 and a standard devaiton of 20. Use the 68-95-99.7 rule to find the following quantities:

a. percentages of scores less than 100
b. relative frequency of scores less than 120
c.percentage of scores less than 140

Here are a few hints:

a. The mean divides a distribution in half: 50% of the distribution is above the mean and 50% of the distribution is below the mean.

b. A score of 120 is one standard deviation above the mean.

c. A score of 140 is two standard deviations above the mean.

I'll let you take it from here.

a. 50%

b. 0.84
c. 97.72%

To find the percentages of scores less than a certain value, we can use the 68-95-99.7 rule, which gives us an approximate idea of how the data is distributed within certain ranges.

a. Percentages of scores less than 100:
According to the 68-95-99.7 rule, approximately 68% of the scores fall within one standard deviation of the mean. Since the mean is 100 and the standard deviation is 20, one standard deviation below the mean is 100 - 20 = 80.

Therefore, the percentage of scores less than 100 is approximately (100 - 80) / 2 = 10%.

b. Relative frequency of scores less than 120:
The 68-95-99.7 rule tells us that approximately 95% of the scores fall within two standard deviations of the mean. In this case, two standard deviations below the mean is 100 - (2 * 20) = 60.

Therefore, the relative frequency of scores less than 120 is approximately (120 - 60) / 2 = 30%.

c. Percentage of scores less than 140:
According to the 68-95-99.7 rule, approximately 99.7% of the scores fall within three standard deviations of the mean. In this case, three standard deviations below the mean is 100 - (3 * 20) = 40.

Therefore, the percentage of scores less than 140 is approximately (140 - 40) / 2 = 50%.

To find the percentages of scores less than a certain value, we can use the 68-95-99.7 rule, also known as the empirical rule, which gives us a rough idea of how the data is distributed in a normal distribution.

a. Percentages of scores less than 100:
According to the 68-95-99.7 rule, approximately 68% of the data falls within 1 standard deviation of the mean. Since the mean is 100 and the standard deviation is 20, 1 standard deviation below the mean is 100 - 20 = 80. Therefore, approximately 68% of scores are less than 100.

b. Relative frequency of scores less than 120:
To find the relative frequency, we need to consider the percentage of scores that fall within 2 standard deviations of the mean. According to the 68-95-99.7 rule, approximately 95% of the data falls within this range. So, we take 95% - 68% = 27% to find the additional percentage of scores that fall between 1 and 2 standard deviations above the mean.

Since 2 standard deviations above the mean is 100 + 2 * 20 = 140, we know that approximately 95% + 27% = 122% of scores are less than 140. However, since percentages cannot be greater than 100%, we can say that the relative frequency of scores less than 120 is 100%.

c. Percentage of scores less than 140:
Since we already established that approximately 122% of scores are less than 140, we can conclude that 100% of the scores are less than 140.