Assume that a set of test scores is normally distrbuted 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.
Z = (x - mean)/SD
mean ± 1 SD = 68%
mean ± 2 SD = 95%
mean ± 3 SD = 99%
a. In a normal distribution, mean= median. What does that tell you?
b. 120 = mean + 1 SD
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Students were given an exam with 300 multiple-choice questions. The distribution of the scores was normal and mean was 195 with a standard deviation of 30. What were the scores of the students who were within one standard deviation of the mean?
To answer these questions, we can use the 68-95-99.7 rule, also known as the Empirical Rule or the Three Sigma Rule. This 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.
Now let's apply this rule to the given scenario:
a. Percentage of scores less than 100:
Since the mean is 100, this is the same as finding the percentage of scores within one standard deviation of the mean, which is 68%.
b. Relative frequency of scores less than 120:
To find this, we need to calculate the percentage of scores within two standard deviations of the mean. Since the standard deviation is 20, two standard deviations would be 40. So, the relative frequency within two standard deviations is approximately 95%.
c. Percentage of scores less than 140:
Here, we want to find the percentage of scores within three standard deviations of the mean, which is approximately 99.7%.
d. Percentage of scores less than 80:
Since 80 is 20 units below the mean, and the standard deviation is 20, we can say that the percentage of scores less than 80 is the same as the percentage of scores within one standard deviation below the mean, which is 16% (68% - 34%).
e. Relative frequency of scores less than 60:
To find this, we need to calculate the percentage of scores within two standard deviations below the mean. Since 60 is 40 units below the mean, and the standard deviation is 20, two standard deviations below the mean is 80. So, the relative frequency within two standard deviations below the mean is approximately 2.5% (95% - 2.5%).
f. Percentage of scores greater than 120:
Since we know that approximately 95% of the data falls within two standard deviations of the mean, and we want to find the percentage of scores greater than 120, we can subtract this percentage from 100% to get the percentage of scores greater than 120. So, it would be approximately 5%.
It's important to note that these percentages are approximations based on the 68-95-99.7 rule, which assumes a normal distribution. In practice, the actual percentages may deviate slightly from these values.