Question
When Marques commutes to work, the amount of time it takes him to arrive is normally
distributed with a mean of 37 minutes and a standard deviation of 4.5 minutes. Using the
empirical rule, what percentage of his commutes will be between 23.5 and 50.5 minutes?
Answer Attempt 1 out of 2
distributed with a mean of 37 minutes and a standard deviation of 4.5 minutes. Using the
empirical rule, what percentage of his commutes will be between 23.5 and 50.5 minutes?
Answer Attempt 1 out of 2
Ask by Bates Gibson. in the United States
Jan 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
About 99.7% of Marques’s commutes will be between 23.5 and 50.5 minutes.
Solution
To determine the percentage of Marques’s commutes that fall between 23.5 and 50.5 minutes using the empirical rule, let’s follow these steps:
1. Understand the Empirical Rule
The empirical rule, also known as the 68-95-99.7 rule, applies to normal distributions and states that:
- 68% of the data falls within ±1 standard deviation from the mean.
- 95% within ±2 standard deviations.
- 99.7% within ±3 standard deviations.
2. Identify the Given Values
- Mean (μ): 37 minutes
- Standard Deviation (σ): 4.5 minutes
- Interval: 23.5 to 50.5 minutes
3. Calculate the Number of Standard Deviations
First, determine how many standard deviations the interval limits are from the mean.
-
Lower Limit: 23.5 minutes
-
Upper Limit: 50.5 minutes
Both limits are 3 standard deviations away from the mean.
4. Apply the Empirical Rule
Since the interval from 23.5 to 50.5 minutes spans ±3 standard deviations from the mean:
- Percentage of Commutes Within This Range: 99.7%
Conclusion
Approximately 99.7% of Marques’s commutes will take between 23.5 and 50.5 minutes based on the empirical rule.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
Using the empirical rule, also known as the 68-95-99.7 rule, we can analyze Marques’ commute times. The mean is 37 minutes, and the standard deviation is 4.5 minutes. According to the rule, approximately 68% of the data fall within one standard deviation (32.5 to 41.5 minutes), about 95% within two standard deviations (28 to 46 minutes), and roughly 99.7% within three standard deviations (23.5 to 50.5 minutes). So, you can expect around 99.7% of his commutes to fall between those two times!
Understanding how the empirical rule works can also help you become a data-savvy commuter! If you know the average time and how much it varies, you can better plan your schedule. For instance, if you’re looking to arrive on time consistently, seeing that most of Marques’ commutes are within 23.5 and 50.5 minutes gives you a pretty good cushion when setting out for work!
