Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The accompanying data were obtained in a study conducted by the manager of SavMore Supermarket. In this study, the number of customers waiting in line at the express checkout at the beginning of each 3 -min interval between 9 a.m. and 12 noon on Saturday was observed. $$\begin{array}{llllll}\hline \text { Customers } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Frequency of } & & & & & \\ \text { Occurrence } & 1 & 4 & 2 & 7 & 14 \\ \hline\end{array}$$ $$\begin{array}{lllllll}\hline \text { Customers } & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline \text { Frequency of } & & & & & & \\\\\text { Occurrence } & 8 & 10 & 6 & 3 & 4 & 1 \\\\\hline\end{array}$$ a. Find the probability distribution of the random variable \(X\), where \(X\) denotes the number of customers observed waiting in line. b. Draw the histogram representing the probability distribution.

Step-by-Step Solution

Answer

The probability distribution of the random variable \(X\), where \(X\) denotes the number of customers observed waiting in line, is as follows: $$ \begin{array}{lllllll} \hline \text { Customers } & 0 & 1 & & 2 & & 3 & 4 \\\ \hline \text { Probability } & \frac{1}{28} & \frac{4}{28} & \frac{2}{28} & \frac{7}{28} & \frac{14}{28} \\\ \hline \end{array} $$ The histogram should have the number of customers on the x-axis and the probability on the y-axis. The width of each bar should be the same, and the height of each bar should correspond to the probability for that number of customers.

Step 1: Calculate the total number of observations

Add up the frequency of occurrence for each possible number of customers waiting in line: Total number of observations = 1 + 4 + 2 + 7 + 14 = 28

Step 2: Calculate the probability of each possible number of customers

Divide the frequency of occurrence for each number of customers by the total number of observations: P(0) = 1/28 P(1) = 4/28 P(2) = 2/28 P(3) = 7/28 P(4) = 14/28

Step 3: Create a probability distribution table

Organize the probabilities for each possible number of customers in a table: $$ \begin{array}{lllllll} \hline \text { Customers } & 0 & 1 & & 2 & & 3 & 4 \\\ \hline \text { Probability } & \frac{1}{28} & \frac{4}{28} & \frac{2}{28} & \frac{7}{28} & \frac{14}{28} \\\ \hline \end{array} $$

Step 4: Draw the histogram

Create a histogram with the number of customers on the x-axis and the probability on the y-axis. The width of each bar should be the same, and the height of each bar should correspond to the probability for that number of customers. [Insert histogram here.]

Key Concepts

Random Variable

In statistics, a random variable is a crucial concept that represents the possible outcomes of a statistical experiment. It assigns numerical values to different outcomes in a sample space. For instance, in the context of SavMore Supermarket's study, the random variable, denoted by \(X\), represents the number of customers waiting in the checkout line at various time intervals.

A random variable can be discrete or continuous. A discrete random variable, like \(X\) in this study, can take on a finite or countably infinite number of values, such as 0, 1, 2, up to 10 customers. In contrast, a continuous random variable can take on an infinite number of values within a given range.

Understanding random variables allows us to mathematically describe and analyze real-world phenomena by calculating probabilities and making predictions.

Histogram

A histogram is a type of bar graph that represents the frequency of different numerical outcomes in a dataset. It's commonly used in statistics to provide a visual summary of a data distribution. In the case of the SavMore Supermarket study, a histogram can help us understand how often different numbers of customers are observed waiting in line.

Each bar in a histogram represents a range of values (in this case, the number of customers). The height of each bar corresponds to the frequency or probability of those values occurring. To draw a histogram for the supermarket data, plot the number of customers on the x-axis and the corresponding probabilities on the y-axis.

By observing the histogram, students can easily identify the most common number of customers and understand the shape of the probability distribution, such as whether it is skewed or symmetrical.

Frequency Distribution

Frequency distribution is a way to organize data to show how often each possible outcome occurs. In the supermarket study, we have a frequency distribution that tells us how many times we observe a specific number of customers waiting in line during the three-minute intervals.

This distribution can be displayed through tables or graphs and is a foundational tool in statistics that helps us understand data patterns. It allows us to quickly assess which outcomes are most and least frequent. For example, in SavMore's data set, 4 customers waiting seem the most common, followed by 6, according to their frequencies.

From the frequency distribution, we can calculate probabilities for each outcome, leading to the probability distribution, an essential step in interpreting data and making informed decisions.