- Calculate the \(z\)-scores using the formula \(z = \frac{X - \mu}{\sigma}\) with \(\mu = 25\) and \(\sigma = 6.1\).
- For \(X = 42\), \(z = 2.79\), giving \(P(X > 42) = 1 - P(Z \leq 2.79) \approx \boxed{0.0026}\).
- For \(X = 8\), \(z = -2.79\), giving \(P(X < 8) \approx \boxed{0.0026}\).
Explanation
- Analyze the distribution and the z-score formula
We are given that the commuting time \(X\) follows a normal distribution \(X \sim N(\mu, \sigma^2)\) with mean \(\mu = 25\) minutes and standard deviation \(\sigma = 6.1\) minutes. To find probabilities for this normal distribution, we convert the raw scores \(X\) into standard normal \(z\)-scores using the formula:
\[z = \frac{X - \mu}{\sigma}\]
where \(z\) represents the number of standard deviations a value \(X\) is from the mean. We will round these \(z\)-scores to two decimal places as requested.
- Calculate the probability for more than 42 minutes
To find the probability that a commute is more than 42 minutes, i.e., \(P(X > 42)\), we first calculate the \(z\)-score for \(X = 42\):
\[z_a = \frac{42 - 25}{6.1} = \frac{17}{6.1} \approx 2.78688...\]
Rounding to two decimal places, we get \(z_a = 2.79\).
Using the standard normal distribution, the probability \(P(Z > 2.79) = 1 - P(Z \leq 2.79)\). Consulting standard normal tables or statistical software, we find \(P(Z \leq 2.79) \approx 0.9974\).
Therefore, \(P(X > 42) \approx 1 - 0.99736 = 0.0026\).
- Calculate the probability for less than 8 minutes
To find the probability that a commute is less than 8 minutes, i.e., \(P(X < 8)\), we calculate the \(z\)-score for \(X = 8\):
\[z_b = \frac{8 - 25}{6.1} = \frac{-17}{6.1} \approx -2.78688...\]
Rounding to two decimal places, we get \(z_b = -2.79\).
Using the standard normal distribution, the probability \(P(Z < -2.79) \approx 0.0026\).
The symmetry of the normal distribution means that \(P(Z < -z) = P(Z > z)\), which confirms our result is identical to the first part due to the symmetric distance of 42 and 8 from the mean of 25.
- Final conclusion and summary of probabilities
Combining our results, we have determined that for a normally distributed commute time with \(\mu = 25\) and \(\sigma = 6.1\):
AThe probability of a commute being more than 42 minutes is \(P(X > 42) \approx 0.0026\).
BThe probability of a commute being less than 8 minutes is \(P(X < 8) \approx 0.0026\).
Examples
This type of analysis is crucial in urban planning and transit management. For instance, if a transit agency wants to know how many commuters will be significantly late, they can use the normal distribution to estimate the percentage of travelers exceeding a certain time threshold. Similarly, it helps businesses estimate how early employees might arrive, allowing them to optimize their office hours or support systems. \(\) Calculate the probability of a value falling beyond a threshold using the z-score: \(z = \frac{X - \mu}{\sigma}\). If \(\mu = 25\) and \(\sigma = 6.1\), to find \(P(X > 42)\), compute \(z = \frac{42 - 25}{6.1} = 2.79\). The probability is then found using the cumulative distribution function.