Question

Create a dotplot for the number of cheeseburgers eaten in a given week by 10 college students. \(\begin{array}{llll}4 & 5 & 4 & 2\end{array}\) \(\begin{array}{lllll}3 & 3 & 4 & 2 & 7\end{array}\) a. How would you describe the shape of the distribution? b. What proportion of the students ate more than 4 cheeseburgers that week?

Short answer

Based on the dot plot created for the number of cheeseburgers eaten in a week by 10 college students, describe the shape of the distribution and find the proportion of students who ate more than 4 cheeseburgers in a week.

Step-by-step solution

Combine the datasets

Combine the two given datasets into one: \([4, 5, 4, 2, 3, 3, 4, 2, 7]\)

Create the dot plot

In order to create the dot plot, we will first find the minimum and maximum values in the dataset. Then we will display the data using dots. Minimum value: 2 Maximum value: 7 Create the dot plot. The number of cheeseburgers consumed is represented on the x-axis and the number of students is represented on the y-axis. ``` 7 | * 6 | 5 | * 4 | * * * 3 | * * 2 | * * |________ 1 2 3 4 5 6 7 ```

Describe the shape of the distribution

In this dot plot, there is a peak around the value of 4, indicating the highest concentration of the data. The shape of the distribution looks slightly skewed to the right since there are more data points to the right of the peak.

Calculate the proportion of students who ate more than 4 cheeseburgers in a week

To find the proportion of students who ate more than 4 cheeseburgers in a week, first count the number of students that fit this criterion (ate more than 4 cheeseburgers) and divide by the total number of students. Number of students who ate more than 4 cheeseburgers: 2 (5 and 7 cheeseburgers) Total number of students: 9 Proportion of students who ate more than 4 cheeseburgers in a week: \(\frac{2}{9}\) Answer: a. The distribution is slightly skewed to the right. b. The proportion of students who ate more than 4 cheeseburgers in a week is \(\frac{2}{9}\).

Key concepts

Data Visualization

Data visualization is a key component in understanding and communicating statistical information effectively. In this exercise, we created a dotplot, which is one of the simplest forms of data visualization, yet very powerful. It involves placing dots along an axis to represent the frequency or count of a particular value in a dataset.

In our case, the dotplot helps to visually convey the number of cheeseburgers eaten by a small group of college students over the course of a week. The horizontal axis (X-axis) represents the possible number of cheeseburgers consumed, while the vertical stacking of dots represents the number of students who ate that quantity. This display makes it easier to see patterns, such as clusters of data, and anomalies like outliers. Dotplots are particularly helpful for small datasets, providing a clear picture of the data distribution at a glance.

When students are constructing dotplots, they should ensure that each category (number of cheeseburgers) is spaced equally along the axis, and each dot represents one observation in the data. In our textbook exercise, students might improve their understanding of the concept by physically plotting dots on graph paper or using software tools designed for statistical graphs.

Descriptive Statistics

Descriptive statistics involve summarizing and organizing the information in a dataset to highlight its key characteristics. One major aspect is describing the shape of the distribution, which tells us how the data is spread out. In our exercise, the distribution is described as slightly right-skewed, meaning that the majority of the students ate fewer than 4 cheeseburgers, with a smaller number eating more.

Another part of descriptive statistics is identifying the measure of central tendency, such as the mean or median, which represents a 'typical' value. Even though our exercise doesn't calculate these, incorporating such measures could further student understanding. Additionally, understanding variability through the range (maximum value minus minimum value) or standard deviation can provide insights into the consistency of students' cheeseburger-eating habits.

Describing data distributions aids in making informed decisions and predictions. For example, a fast-food restaurant near the campus could use such statistics to estimate demand for cheeseburgers during the week.

Probability and Statistics

Probability and statistics are intertwined, with probability providing the theoretical foundation for statistical analysis. The dotplot exercise touches upon probability when we calculate the proportion of students who ate more than 4 cheeseburgers. This proportion is essentially an empirical probability, estimating the likelihood of an event (eating more than 4 cheeseburgers) based on observed data.

In the exercise, finding that 2 out of 9 students ate more than 4 cheeseburgers sets the stage for calculating the probability of a randomly selected student exceeding this threshold. By expressing this as the ratio \(\frac{2}{9}\), we venture into probability concepts, understanding that not all events are equally likely.

Teaching students to connect descriptive statistics with probability helps them to anticipate outcomes and assess risks in real-world situations. For instance, the probability calculated from the students' eating habits could suggest the likelihood of a similar consumption pattern in comparable populations.