The mean on a Advanced Algebra test was 78 with a standard deviation of 8. If the test scores are normal distributed, find the interval about the mean that contains 99.7% of the scores. Use the empirical rule.
To find the interval about the mean that contains 99.7% of the scores using the empirical rule, we can use the properties of a normal distribution.
The empirical 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 that the mean is 78 and the standard deviation is 8, we can use the empirical rule to find the interval.
1. Within one standard deviation:
The interval will be from 78 - 8 to 78 + 8, which is from 70 to 86.
2. Within two standard deviations:
The interval will be from 78 - (2 * 8) to 78 + (2 * 8), which is from 62 to 94.
3. Within three standard deviations:
The interval will be from 78 - (3 * 8) to 78 + (3 * 8), which is from 54 to 102.
Therefore, the interval about the mean that contains 99.7% of the scores is from 54 to 102.
To find the interval about the mean that contains 99.7% of the scores, we can use the empirical rule, also known as the 68-95-99.7 rule.
According to the empirical rule, for a normal distribution:
- About 68% of the data falls within one standard deviation of the mean.
- About 95% of the data falls within two standard deviations of the mean.
- About 99.7% of the data falls within three standard deviations of the mean.
Given that the mean is 78 and the standard deviation is 8, we can use these percentages to determine the interval about the mean.
First, we calculate one standard deviation below and above the mean:
1. Lower bound: 78 - (1 * 8) = 70
2. Upper bound: 78 + (1 * 8) = 86
Next, we calculate two standard deviations below and above the mean:
1. Lower bound: 78 - (2 * 8) = 62
2. Upper bound: 78 + (2 * 8) = 94
Finally, we calculate three standard deviations below and above the mean:
1. Lower bound: 78 - (3 * 8) = 54
2. Upper bound: 78 + (3 * 8) = 102
Therefore, the interval about the mean that contains 99.7% of the scores is from 54 to 102.
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.
±3Z = (score-mean)/SD