Using the 68-95-99.7rule. 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 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

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To answer these questions using the 68-95-99.7 rule, we need to understand how it applies to a normal distribution. The 68-95-99.7 rule states that for a normally distributed data set:

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

a. Percentage of scores less than 100:
Since the mean is 100 and the scores are normally distributed, we know that 50% of the scores are below the mean. Additionally, due to the symmetry of the normal distribution, the remaining 50% of the scores are above the mean. Therefore, the percentage of scores less than 100 is 50%.

b. Relative frequency of scores less than 120:
To find the relative frequency, we need to determine how many standard deviations above the mean a score of 120 is. Taking the mean as our starting point (which is 100), we calculate the difference between the score and the mean: 120 - 100 = 20. Since the standard deviation is 20, this means that the score of 120 is one standard deviation above the mean. According to the 68-95-99.7 rule, approximately 68% of the scores fall within one standard deviation of the mean. Therefore, the relative frequency of scores less than 120 is approximately 68%.

c. Percentage of scores less than 140:
Just like in part b, we calculate the difference between the score of 140 and the mean of 100: 140 - 100 = 40. Since the standard deviation is 20, this means the score of 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, 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 calculate the difference between the score of 80 and the mean of 100: 80 - 100 = -20. Since the standard deviation is 20, this means that 80 is one standard deviation below the mean. Again using the 68-95-99.7 rule, approximately 68% of the scores fall within one standard deviation of the mean. Therefore, the percentage of scores less than 80 is approximately 68%.

e. Relative frequency of scores less than 60:
To find the relative frequency, we calculate the difference between the score of 60 and the mean of 100: 60 - 100 = -40. Since the standard deviation is 20, this means that 60 is two 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, the relative frequency of scores less than 60 is approximately 95%.

f. Percentage of scores greater than 120:
Using the 68-95-99.7 rule, we know that approximately 68% of scores fall within one standard deviation of the mean, which means that approximately 32% of scores fall beyond one standard deviation away from the mean on both sides. Since the distribution is symmetrical, this implies that approximately 16% of scores fall above one standard deviation above the mean. Therefore, the percentage of scores greater than 120 is approximately 16%.