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:
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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 rule? For example, 68% are with in one standard deviation of the mean in both directions, 95% are within 2 SD and 99.7% are within 3 SD.
a. in a normal distribution, mean = median. What does that tell you?
b. Mean + 1 SD = ?
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To answer these questions using the 68-95-99.7 rule, also known as the empirical rule, we need to understand the concept of standard deviations and how they relate to percentages.
The 68-95-99.7 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 find the answers to the questions:
a. Percentage of scores less than 100:
Since the mean is 100 and the distribution is normal, we know that 50% of the scores will be below the mean. Therefore, 50% of the scores will be above the mean. So, the percentage of scores less than 100 will be 50%.
b. Relative frequency of scores less than 120:
We need to find the percentage of scores that fall within one standard deviation of the mean. Since the standard deviation is given as 20, one standard deviation above the mean is at 100 + 20 = 120. Using the 68% rule, approximately 68% of the scores will be less than 120.
c. Percentage of scores less than 140:
Similarly to the previous question, we need to find the percentage of scores that fall within two standard deviations of the mean. Since two standard deviations above the mean is at 100 + 2*20 = 140, using the 95% rule, approximately 95% of the scores will be less than 140.
d. Percentage of scores less than 80:
Using the 68% rule, we know that approximately 68% of the scores will be within one standard deviation of the mean. Since one standard deviation below the mean is at 100 - 20 = 80, the percentage of scores less than 80 is also approximately 16%. (half of the 32% outside the first standard deviation, since the distribution is symmetric)
e. Relative frequency of scores less than 60:
Again using the 68% rule, we know that approximately 68% of the scores will be within one standard deviation of the mean. Since two standard deviations below the mean is at 100 - 2*20 = 60, the relative frequency (in this case, the percentage) of scores less than 60 is also approximately 2.5%. (half of the 5% outside the second standard deviation, since the distribution is symmetric)
f. Percentage of scores greater than 120:
This can be calculated by subtracting the percentage of scores less than 120 from 100%. As we calculated in question (b), approximately 68% of the scores will be less than 120. Therefore, approximately 100% - 68% = 32% of the scores will be greater than 120.
To summarize:
a. Percentage of scores less than 100: 50%
b. Relative frequency of scores less than 120: 68%
c. Percentage of scores less than 140: 95%
d. Percentage of scores less than 80: 16%
e. Relative frequency of scores less than 60: 2.5%
f. Percentage of scores greater than 120: 32%