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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?

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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Interpret measures of center and variability

Full solution

Q. 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?

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