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:
a. Percentage of scores less than 100=50 percent
b. Relative frequency of scores less than 120= 0.34+0.135+0.0235+0.015+0.84
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
I believe Statistics is your school subject.
To find the required quantities using the 68-95-99.7 rule, we need to understand the standard deviation and its relationship to the normal distribution.
The 68-95-99.7 rule, also known as the empirical 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
- Approximately 99.7% falls within three standard deviations
Using this rule, we can calculate the quantities:
a. Percentage of scores less than 100:
Since the mean is 100, this falls within the first standard deviation. Thus, approximately 68% of the scores are less than 100.
b. Relative frequency of scores less than 120:
Since 120 is one standard deviation above the mean, it falls within the second standard deviation. This corresponds to 95% - 68% = 27% of the scores. However, we need to be careful with the wording of the question, which asks for the relative frequency. So we sum up the percentages for the first and second standard deviations: 68% + 27% = 0.68 + 0.27 = 0.95. Therefore, the relative frequency of scores less than 120 is 0.95 or 95%.
c. Percentage of scores less than 140:
Since 140 is two standard deviations above the mean, it falls within the third standard deviation. This corresponds to 99.7% - 95% = 4.7% of the scores. Therefore, the percentage of scores less than 140 is approximately 95% + 4.7% = 99.7%.
d. Percentage of scores less than 80:
Since 80 is one standard deviation below the mean, it falls within the first standard deviation. Thus, approximately 68% of the scores are less than 80.
e. Relative frequency of scores less than 60:
Since 60 is two standard deviations below the mean, it falls within the third standard deviation. This corresponds to 99.7% - 95% = 4.7% of the scores. Therefore, the relative frequency of scores less than 60 is approximately 0.047 or 4.7%.
f. Percentage of scores greater than 120:
To find the percentage of scores greater than 120, we subtract the percentage of scores less than 120 from 100%. From part (b), we know that the relative frequency of scores less than 120 is 0.95 or 95%. Therefore, the percentage of scores greater than 120 is approximately 100% - 95% = 5%.