Problems 11–14 use the information presented in Examples 1 and 2.  
b. It is 10 A.M. There is a 90% probability your friend will arrive within the next _______ minutes.

Verified step by step guidance

1. Identify the type of probability distribution that models your friend's arrival time. Typically, arrival times within a certain interval can be modeled using a uniform or normal distribution depending on the context given in Examples 1 and 2.
2. Determine the parameters of the distribution from the examples, such as the mean (\$\mu\$) and standard deviation (\$\sigma\$) if it is a normal distribution, or the interval limits if it is uniform.
3. Since it is currently 10 A.M., define the random variable \$X\$ as the time (in minutes) after 10 A.M. that your friend arrives. You want to find the time \$t\$ such that the probability \$P(X \leq t) = 0.90\$.
4. Use the cumulative distribution function (CDF) of the distribution to set up the equation \$P(X \leq t) = 0.90\$. For a normal distribution, this involves finding the z-score \$z\$ such that \$\Phi(z) = 0.90\$, where \$\Phi\$ is the standard normal CDF, and then converting \$z\$ back to the original scale using \$t = \mu + z \sigma\$.
5. Solve for \$t\$ to find the number of minutes after 10 A.M. within which there is a 90% probability your friend will arrive.

Key Concepts

Probability and Probability Distributions

Probability measures the likelihood of an event occurring, ranging from 0 to 1. Probability distributions describe how probabilities are assigned to different outcomes, often represented by functions like the normal or exponential distribution, which help calculate the chance of events within specific intervals.

Confidence Intervals and Percentiles

A confidence interval or percentile indicates a range within which a certain percentage of data points lie. For example, a 90% probability corresponds to the 90th percentile, meaning there is a 90% chance the event occurs within that range, which is essential for determining time intervals in arrival problems.

Using Examples and Given Data in Problem Solving

Applying information from provided examples or data sets is crucial for solving related problems. This involves interpreting given parameters, such as mean arrival times or distribution types, and using them to calculate probabilities or time intervals relevant to the question.