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When Aaliyah commutes to work, the amount of time it takes her to arrive is normally distributed with a mean of 21 minutes and a standard deviation of 2 minutes.
Out of the 284 days that Aaliyah commutes to work per year, how many times would her commute be shorter than 17 minutes, to the nearest whole number?
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.
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