When Michael commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 24 minutes and a standard deviation of 3 minutes. Out of the 273 days that Michael commutes to work per year, how many times would his commute be between 19 and 31 minutes, to the nearest whole number?

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Identify Parameters: Identify the parameters of the normal distribution. The mean (μ) is 24 minutes, and the standard deviation (σ) is 3 minutes.Convert to Z-scores: Convert the commute time range to z-scores. To find the z-score for 19 minutes: $ z = \frac{X - \mu}{\sigma} = \frac{19 - 24}{3} = \frac{-5}{3} \approx -1.67 $ To find the z-score for 31 minutes: $ z = \frac{X - \mu}{\sigma} = \frac{31 - 24}{3} = \frac{7}{3} \approx 2.33 $Use Normal Distribution Table: Use the standard normal distribution table to find the probabilities corresponding to the z-scores. The probability of a z-score being less than -1.67 is approximately 0.0475. The probability of a z-score being less than 2.33 is approximately 0.9901.Calculate Probability Range: Calculate the probability of the commute time being between 19 and 31 minutes. Subtract the probability of the commute being less than 19 minutes from the probability of the commute being less than 31 minutes. Probability between 19 and 31 minutes = 0.9901 - 0.0475 = 0.9426Calculate Number of Days: Calculate the number of days Michael's commute would be between 19 and 31 minutes. Multiply the total number of commuting days by the probability found in Step 4. Number of days = 273 * 0.9426 ≈ 257.4Round to Nearest Whole Number: Round the result to the nearest whole number. The number of days Michael's commute would be between 19 and 31 minutes is approximately 257 days when rounded to the nearest whole number.

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