To solve this problem using the empirical rule (68-95-99.7 rule), let’s analyze the given information:

• Mean ($\mu$) = 37 minutes
• Standard deviation ($\sigma$) = 4.5 minutes

The empirical rule states that for a normal distribution:

1. Approximately 68% of the data falls within 1 standard deviation of the mean ($\mu \pm \sigma$).
2. Approximately 95% of the data falls within 2 standard deviations of the mean ($\mu \pm 2\sigma$).
3. Approximately 99.7% of the data falls within 3 standard deviations of the mean ($\mu \pm 3\sigma$).

Step 1: Calculate $\mu \pm \sigma$ and $\mu \pm 2\sigma$

• $\mu - \sigma = 37 - 4.5 = 32.5$

• $\mu + \sigma = 37 + 4.5 = 41.5$

• $\mu - 2\sigma = 37 - 2(4.5) = 28$

• $\mu + 2\sigma = 37 + 2(4.5) = 46$

Step 2: Determine the interval 23.5 to 50.5

• The interval $23.5$ to $50.5$ spans from $\mu - 3\sigma$ to $\mu + 3\sigma$:
  • $\mu - 3\sigma = 37 - 3(4.5) = 23.5$
  • $\mu + 3\sigma = 37 + 3(4.5) = 50.5$

Step 3: Apply the empirical rule

The interval $23.5$ to $50.5$ corresponds to $\mu \pm 3\sigma$, which, according to the empirical rule, includes 99.7% of the commutes.

Final Answer:

99.7% of Marques' commutes will be between 23.5 and 50.5 minutes.

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Would you like a deeper breakdown or additional examples?

Related Questions:

1. What percentage of commutes are shorter than 23.5 minutes?
2. What is the probability a commute is between 32.5 and 41.5 minutes?
3. Can you explain how to use the Z-score to verify this result?
4. How would the percentages change if the standard deviation were 6 minutes?
5. What are the limits of the empirical rule for real-world applications?

Tip:

When using the empirical rule, always start by identifying how many standard deviations the interval spans to quickly approximate probabilities.