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:
Suggest you make a drawing and label first…Drawings need not be included in response.
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
Make the drawing.
a. Remember that the mean = median in a normal distribution.
b. Since 68% score within one SD of the mean, 34% score above the mean and 34% score below. Since you are at +1 SD, add % below mean and the 34% between mean and +1 SD.
Use a similar process for answering the remaining problems.
To use the 68-95-99.7 rule to find the percentages and relative frequencies, we need to understand the concept of standard deviation and the properties of a normal distribution.
1. Start by drawing a normal distribution curve on a graph or paper. Label the x-axis as "Test Scores" and the y-axis as "Probability Density".
2. Plot the mean of 100 on the x-axis. This represents the center of the distribution.
3. Since the standard deviation is 20, mark one standard deviation above and below the mean on the graph. This would be 80 and 120.
Now, let's calculate the percentages and relative frequencies:
a. Percentage of scores less than 100:
Since the mean is 100, any score less than 100 would be to the left of the mean. According to the 68-95-99.7 rule, approximately 50% of the scores fall to the left of the mean. Therefore, the percentage of scores less than 100 is 50%.
b. Relative frequency of scores less than 120:
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 68%.
c. Percentage of scores less than 140:
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 95%.
d. Percentage of scores less than 80:
80 is one standard deviation below the mean. According to the 68-95-99.7 rule, approximately 68% of the scores fall within one standard deviation of the mean. Since we want the percentage of scores less than 80, we can subtract 68% from 50% (which represents the scores to the left of the mean). Therefore, the percentage of scores less than 80 is approximately 32%.
e. Relative frequency of scores less than 60:
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. Since we want the relative frequency of scores less than 60, we can subtract 95% from 50% (which represents the scores to the left of the mean). Therefore, the relative frequency of scores less than 60 is approximately 45%.
f. Percentage of scores greater than 120:
Since we know that approximately 68% of the scores fall within one standard deviation of the mean (between 80 and 120), we can subtract 68% from 100% to find the percentage of scores greater than 120. Therefore, the percentage of scores greater than 120 is approximately 32%.
Remember, these percentages and relative frequencies are approximations based on the 68-95-99.7 rule and assume a normal distribution.