Sure, let’s break it down step by step to find the solution to the problem:
1. Identify the Given Information:
- Mean commuting time ([tex]\(\mu\)[/tex]) = 21 minutes
- Standard deviation ([tex]\(\sigma\)[/tex]) = 2 minutes
- Number of commuting days in a year = 284 days
- Target commuting time limit = 17 minutes
2. Calculate the Z-Score for the Target Time:
The Z-score represents the number of standard deviations a data point (in this case, 17 minutes) is from the mean. The formula for the Z-score is:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
Where [tex]\(X\)[/tex] is the target time (17 minutes), [tex]\(\mu\)[/tex] is the mean (21 minutes), and [tex]\(\sigma\)[/tex] is the standard deviation (2 minutes).
[tex]\[
Z = \frac{17 - 21}{2} = \frac{-4}{2} = -2.0
\][/tex]
3. Find the Probability Corresponding to the Z-Score:
Using the Z-score of -2.0, we consult the standard normal distribution table (or use statistical software) to find the cumulative probability associated with this Z-score. The cumulative probability tells us the likelihood that a value is less than the target value.
The cumulative probability for a Z-score of -2.0 is approximately 0.02275.
4. Calculate the Expected Number of Days with Commute Shorter than the Target Time:
To find the expected number of days that Aaliyah’s commute will be shorter than 17 minutes, multiply the total number of commuting days by the cumulative probability:
[tex]\[
\text{Expected Days} = \text{Total Commuting Days} \times \text{Probability}
\][/tex]
[tex]\[
\text{Expected Days} = 284 \times 0.02275 \approx 6.46
\][/tex]
5. Round to the Nearest Whole Number:
Since the question asks for the number of times to the nearest whole number, we round 6.46 to 6.
Thus, based on this analysis, Aaliyah would have a commute shorter than 17 minutes approximately 6 times per year.
