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…
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
Do you know the 68-95-99.7 rule? Approximately 68% of scores in normal distribution are within one standard deviation (34% on each side of the mean), 95% within 2 SD, and 99.7% within 3 SD.
a. Mean = median in normal distribution, so that should tell you % less < mean.
b. Z = number of SD above or below mean = (score-mean)/SD
This should start you out.
The mean weekly earnings of all male workers in a state is $775 and the mean weekly earnings of all female workers in the same state is $658. Suppose we select one sample of 400 male workers and another sample of 300 female workers from this state. What is the mean of the sampling distribution of the difference between the mean weekly earnings of these two samples?
a. According to the 68-95-99.7 rule, approximately 68% of scores fall within one standard deviation of the mean. Since the mean is 100 and the standard deviation is 20, the range of scores within one standard deviation of the mean is from 100 - 20 = 80 to 100 + 20 = 120. Therefore, the percentage of scores less than 100 is 50% (since it includes half of the 68% interval).
b. To find the relative frequency of scores less than 120, we can use the same information as above. Since 120 falls within the interval of the mean plus one standard deviation, which includes approximately 68% of the scores, the relative frequency of scores less than 120 is also 68%.
c. Similarly, 140 falls within two standard deviations from the mean, which includes approximately 95% of the scores according to the 68-95-99.7 rule. Therefore, the percentage of scores less than 140 is 95%.
d. 80 falls within the interval of one standard deviation below the mean, which includes approximately 68% of the scores. Therefore, the percentage of scores less than 80 is 68%.
e. To find the relative frequency of scores less than 60, we need to determine the number of standard deviations below the mean this score falls. Since 60 is 40 points below the mean of 100, and the standard deviation is 20, this score is 2 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 also 95%.
f. Since 120 falls within the interval of one standard deviation of the mean, which includes approximately 68% of the scores, the percentage of scores greater than 120 is 32% (100% - 68%).
To use the 68-95-99.7 rule, we need to understand how data is distributed within a normal distribution. The rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Given a mean of 100 and a standard deviation of 20, let's proceed to calculate the quantities:
a. Percentage of scores less than 100:
Since the mean is 100, this means that exactly 50% of the scores are less than 100.
b. Relative frequency of scores less than 120:
To find the relative frequency, we need to determine the percentage of scores that fall within one standard deviation of the mean. In this case, the score is 120.
To calculate this, we subtract the percentage of scores less than 120 (which is 50%) from the percentage of scores greater than 120 (which is 50%). So, the relative frequency of scores less than 120 is 50% - 50% = 0%.
c. Percentage of scores less than 140:
Now we need to determine the percentage of scores that fall within two standard deviations of the mean, which is 140 in this case.
Using the 68-95-99.7 rule, we can estimate that approximately 95% of the scores are less than two standard deviations above the mean. Therefore, the percentage of scores less than 140 is approximately 95%.
d. Percentage of scores less than 80:
Similar to the previous steps, we need to find the percentage of scores that fall within one standard deviation below the mean, which is 80.
Using the 68-95-99.7 rule again, we estimate that approximately 68% of the scores are less than one standard deviation below 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 need to determine the percentage of scores that fall within two standard deviations below the mean, which is 60.
Using the 68-95-99.7 rule, we estimate that approximately 95% of the scores are less than two standard deviations below the mean. Therefore, the relative frequency of scores less than 60 is approximately 95%.
f. Percentage of scores greater than 120:
To find the percentage of scores that are greater than 120, we can use the knowledge that the sum of the percentages on both sides of the mean is 100%.
Since we have already determined that approximately 50% of the scores are less than 120, the percentage of scores greater than 120 is approximately 100% - 50% = 50%.
These calculations can be visualized better on a normal distribution curve, with the mean at the center and the standard deviations marked accordingly.