To determine how many times Aaliyah's commute would be shorter than 17 minutes out of 284 commuting days, we need to follow a systematic approach that involves understanding the properties of the normal distribution.
1. Identify the given parameters:
- Mean of the commute time ([tex]\(\mu\)[/tex]): 21 minutes
- Standard deviation of the commute time ([tex]\(\sigma\)[/tex]): 2 minutes
- Commute threshold: 17 minutes
- Total number of commuting days: 284 days
2. Calculate the z-score for the threshold:
The z-score represents the number of standard deviations a data point is from the mean. It can be calculated using the formula:
[tex]\[
z = \frac{(X - \mu)}{\sigma}
\][/tex]
Where [tex]\(X\)[/tex] is the threshold value (17 minutes in this case).
Plugging in the values:
[tex]\[
z = \frac{(17 - 21)}{2} = \frac{-4}{2} = -2
\][/tex]
3. Determine the probability associated with the z-score:
The next step is to find the cumulative probability for the calculated z-score of -2. This represents the proportion of the commute times that are less than 17 minutes. Using the standard normal distribution table or cumulative distribution function (CDF) for a z-score of -2, we find:
[tex]\[
P(Z < -2) \approx 0.02275
\][/tex]
This means there is approximately a 2.275% chance that Aaliyah's commute time is less than 17 minutes on any given day.
4. Calculate the expected number of days with a commute shorter than 17 minutes:
To find the number of days where Aaliyah's commute is shorter than 17 minutes, multiply the total number of commuting days by the probability:
[tex]\[
\text{Number of days} = 284 \times 0.02275 \approx 6.46
\][/tex]
Since we need this to be a whole number, we round to the nearest whole number:
[tex]\[
\text{Number of days} \approx 6
\][/tex]
Therefore, Aaliyah's commute is expected to be shorter than 17 minutes approximately 6 times out of the 284 days she commutes to work per year.
