When Parker commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 58 minutes and a standard deviation of 4.5 minutes. Using the empirical rule, determine the interval that represents the middle 95% of his commute times.
Question
When Parker commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 58 minutes and a standard deviation of 4.5 minutes. Using the empirical rule, determine the interval that represents the middle 95% of his commute times.
Solution
The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean.
The middle 95% of data falls within 2 standard deviations of the mean.
Given that the mean commute time is 58 minutes and the standard deviation is 4.5 minutes, we can calculate the interval for the middle 95% of commute times as follows:
- Multiply the standard deviation by 2. (4.5 minutes * 2 = 9 minutes)
- Subtract this value from the mean to get the lower end of the interval. (58 minutes - 9 minutes = 49 minutes)
- Add this value to the mean to get the upper end of the interval. (58 minutes + 9 minutes = 67 minutes)
So, the middle 95% of Parker's commute times should fall between 49 and 67 minutes.
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