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 diviation of 20. Use the 68-95-99.7 rult to find the following quantities:
percentage of scores less than 100
relative frequency of scores less than 120.
percentage of scores less than 80.
relative frequency of scores less than 60.
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.
Z = the score in terms of standard deviations = (score-mean)/SD
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 deviation and the properties of a normal distribution.
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.
1. Percentage of scores less than 100:
Since the mean is 100, the percentage of scores less than 100 is equal to the percentage of scores within one standard deviation below the mean. According to the 68-95-99.7 rule, 68% of the data falls within one standard deviation of the mean. This means that approximately 34% of the data is less than 100.
2. Relative frequency of scores less than 120:
To find the relative frequency, we need to determine how many standard deviations away from the mean 120 is.
(120 - 100) / 20 = 1 standard deviation.
According to the 68-95-99.7 rule, approximately 68% of the data falls within one standard deviation of the mean. This means that the relative frequency of scores less than 120 is approximately 68%.
3. Percentage of scores less than 80:
To find this percentage, we need to determine how many standard deviations away from the mean 80 is.
(80 - 100) / 20 = -1 standard deviation.
Since the normal distribution is symmetric, the percentage of scores less than 80 is equal to the percentage of scores greater than (100 - 80) = 20. According to the 68-95-99.7 rule, approximately 16% of the data is greater than one standard deviation away from the mean. Hence, the percentage of scores less than 80 is approximately 16%.
4. Relative frequency of scores less than 60:
To find the relative frequency, we need to determine how many standard deviations away from the mean 60 is.
(60 - 100) / 20 = -2 standard deviations.
According to the 68-95-99.7 rule, approximately 95% of the data falls within two standard deviations of the mean. This means that the relative frequency of scores less than 60 is approximately 95%.
5. Percentage of scores greater than 120:
To find this percentage, we need to determine how many standard deviations away from the mean 120 is.
(120 - 100) / 20 = 1 standard deviation.
Since the normal distribution is symmetric, the percentage of scores greater than 120 is equal to the percentage of scores less than (100 - 120) = -20. According to the 68-95-99.7 rule, approximately 16% of the data is greater than one standard deviation away from the mean. Hence, the percentage of scores greater than 120 is approximately 16%.
Remember, these values are approximations based on the 68-95-99.7 rule and assume a normal distribution.