Problems 11–14 use the information presented in Examples 1 and 2.  
a. Find the probability that your friend is between 15 and 25 minutes late.

Verified step by step guidance

1. Identify the distribution type and parameters from Examples 1 and 2. Typically, problems involving waiting times or lateness are modeled using a normal distribution or an exponential distribution. Confirm which distribution applies and note its mean (\$\mu\$) and standard deviation (\$\sigma\$) if normal, or rate parameter (\$\lambda\$) if exponential.
2. Define the event of interest: your friend being between 15 and 25 minutes late. This corresponds to finding \$P(15 < X < 25)\$, where \$X\$ is the random variable representing the lateness time.
3. If the distribution is normal, standardize the values 15 and 25 using the z-score formula: \$\$z = \frac{X - \mu}{\sigma}\$\$ Calculate \$z_1 = \frac{15 - \mu}{\sigma}\$ and \$z_2 = \frac{25 - \mu}{\sigma}\$.
4. Use the standard normal distribution table or a calculator to find the probabilities corresponding to \$z_1\$ and \$z_2\$: \$\$P(Z < z_2) - P(Z < z_1)\$\$ This difference gives the probability that \$X\$ lies between 15 and 25.
5. If the distribution is exponential, use the cumulative distribution function (CDF) for the exponential distribution: \$\$P(a < X < b) = F(b) - F(a) = (1 - e^{-\lambda b}) - (1 - e^{-\lambda a}) = e^{-\lambda a} - e^{-\lambda b}\$\$ Substitute \$a=15\$ and \$b=25\$ to find the probability.

Key Concepts

Probability of an Interval

This concept involves finding the likelihood that a continuous random variable falls within a specific range, such as between 15 and 25 minutes. It is calculated by determining the area under the probability distribution curve over that interval.

Continuous Probability Distributions

Continuous distributions describe probabilities for variables that can take any value within an interval. Understanding the shape and properties of the distribution (e.g., uniform, normal) is essential to calculate probabilities for time intervals.

Using Examples and Given Data

Referring to provided examples or data sets helps identify the distribution type and parameters (like mean or range). This information is crucial to apply the correct formulas or methods to find the desired probability.