Using the 68-95-99.7 rule:
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
68 percent within one standard deviation from the mean.
b) 50+34=84 percent (within one standard deviation of mean, or 34 percent on each side)
c. 50+95/2=50+47.5=98
d. 50-34
and so forth. Some are called above in percentage, or relative frequency. Relative frequency is percentage divide by 100
if percentage is 14 percent, relative frequency is .14
http://davidmlane.com/hyperstat/z_table.html
To use the 68-95-99.7 rule, also known as the empirical rule, we need to consider the standard deviations (σ) from the mean (μ). The rule states that:
- Approximately 68% of the scores lie within one standard deviation of the mean.
- Approximately 95% of the scores lie within two standard deviations of the mean.
- Approximately 99.7% of the scores lie within three standard deviations of the mean.
a. To find the percentage of scores less than 100, we can use the fact that the mean is 100. Since the score of 100 is the mean, and the rule applies symmetrically around the mean, we can conclude that about 50% of the scores are less than 100.
b. To find the relative frequency of scores less than 120, we need to determine how many standard deviations away from the mean this score is. Using the formula z = (x - μ) / σ, where x is the individual score, μ is the mean, and σ is the standard deviation, we find that z = (120 - 100) / 20 = 1 standard deviation away from the mean. According to the rule, approximately 68% of scores lie within one standard deviation of the mean. Therefore, the relative frequency of scores less than 120 is approximately 68%.
c. Similarly, to find the percentage of scores less than 140, we calculate z = (140 - 100) / 20 = 2 standard deviations away from the mean. According to the rule, approximately 95% of scores lie within two standard deviations of the mean. Therefore, the percentage of scores less than 140 is approximately 95%.
d. To find the percentage of scores less than 80, we calculate z = (80 - 100) / 20 = -1 standard deviation away from the mean. Since the rule applies symmetrically, we know that the percentage of scores less than -1 standard deviation is equal to the percentage of scores greater than +1 standard deviation. According to the rule, the percentage of scores greater than +1 standard deviation is approximately 16%. Therefore, the percentage of scores less than 80 is also approximately 16%.
e. To find the relative frequency of scores less than 60, we calculate z = (60 - 100) / 20 = -2 standard deviations away from the mean. According to the rule, approximately 2.5% of scores lie beyond two standard deviations of the mean on either side. Thus, the relative frequency of scores less than 60 is approximately 2.5%.
f. Finally, to find the percentage of scores greater than 120, we can use the same principle as in part b. Since 120 is one standard deviation away from the mean, approximately 68% of scores lie within one standard deviation of the mean. Therefore, the remaining percentage of scores greater than 120 is approximately 100% - 68% = 32%.
By utilizing the 68-95-99.7 rule, we can estimate these quantities or probabilities without calculating the individual values and percentages.