Question
If the mean and the standard deviation of a continuous random variable that is normally distributed are 30 and 4.3, respectively, find an interval that contains 67% of the distribution
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Answer to a math question If the mean and the standard deviation of a continuous random variable that is normally distributed are 30 and 4.3, respectively, find an interval that contains 67% of the distribution
Gerhard
4.594 Answers
Since we are dealing with a normal distribution, we can use the empirical rule to find the interval that contains 67% of the distribution.
According to the empirical rule, for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Since we want to find an interval that contains 67% of the distribution, we will consider 2 standard deviations on either side of the mean.
The interval that contains 67% of the distribution is:
(30 - 2 \times 4.3, 30 + 2 \times 4.3)
(30 - 8.6, 30 + 8.6)
(21.4, 38.6)
Therefore, the interval that contains 67% of the distribution is $ \boxed{(21.4, 38.6)} $.
According to the empirical rule, for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Since we want to find an interval that contains 67% of the distribution, we will consider 2 standard deviations on either side of the mean.
The interval that contains 67% of the distribution is:
Therefore, the interval that contains 67% of the distribution is $ \boxed{(21.4, 38.6)} $.
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