Question

Cheeseburgers Create a dotplot for the number of cheeseburgers consumed in a given week by 10 college students. \(\begin{array}{lllll}4 & 5 & 4 & 2 & 1\end{array}\) \(\begin{array}{lllll}3 & 3 & 4 & 2 & 7\end{array}\) 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

Answer: The shape of the distribution is relatively symmetric and unimodal, with a peak at the value of 4. 20% of the students ate more than 4 cheeseburgers in a week.

Step-by-step solution

To begin, we will first arrange all the given data points into a single list. This will make it easier for us to create the dot plot and work on the following questions. \(\begin{array}{llllllllll}1 & 2 & 2 & 3 & 3 & 4 & 4 & 4 & 5 & 7\end{array}\) #Step 2: Create the Dot Plot#

Now that we have arranged the data, draw a horizontal axis, and label it from 1 to 7 (the lowest and highest values in our data) and create a vertical axis. For each value in the list, place a dot above the corresponding number on the horizontal axis. If there are multiple data points with the same value, stack the dots vertically for each occurrence. ``` ^ | 3 | ● 2 | ● ● ● ● 1 | ● ● ● ● ● ● +----------------------------> 1 2 3 4 5 6 7 ``` #Step 3: Describe the distribution shape#

Observing the dot plot, we can describe the distribution's shape. The distribution appears to be relatively symmetric, with a peak occurring at the value of 4. This is also a unimodal distribution since it has only one peak. #Step 4: Find the proportion of students who ate more than 4 cheeseburgers#

To calculate the proportion of students who ate more than 4 cheeseburgers, count the number of data points that are greater than 4 (i.e., 5 and 7) and divide it by the total number of students (10). \(\frac{\text{number of students who ate more than 4 cheeseburgers}}{\text{total number of students}} = \frac{2}{10} = 0.2\) So, 20% of the students ate more than 4 cheeseburgers that week.

Key concepts

Dot Plot

A dot plot is a simple way to visualize numerical data. Imagine drawing a number line, which is a horizontal axis, and then marking each data point as a dot above the appropriate number.
Dot plots are particularly useful when you have a small dataset, like the number of cheeseburgers consumed here by ten students.
After listing our dataset in order, we identify each unique value along the horizontal axis, ranging from 1 to 7 in this case. Each time a value appears in our dataset, a dot is added above the corresponding number.

• If a number is repeated, stack dots vertically to show frequency.
• Dot plots display the data's distribution clearly by showing every individual data point.
• They are great for small sets of data because they are simple and easy to read.

Using a dot plot, it's easy to see which number of cheeseburgers was eaten most frequently and identify any patterns.

Distribution Shape

The shape of a distribution tells you about how the data points are spread out. It's like looking at the overall form of the data on your plot. In the case of the distribution of cheeseburgers eaten, we have a unimodal distribution.
This means there is one peak or most frequent value in the dataset, which is at 4 cheeseburgers here.

• A symmetric distribution means the left and right sides of the plot mirror each other. This can be seen when the number of dot occurrences climbs to a peak then descends in a similar fashion.
• Our dot plot is slightly skewed with a higher frequency around 3 and 4.
• While not perfect, our plot generally mirrors itself around the central peak.

Understanding the shape can help us to predict or describe future data points and provides insights into how the data behaves.

Proportion Calculation

Calculating proportions helps us understand parts of the data in relation to the whole. It is a division of one part by the total. For the cheeseburger example, we calculate how many students ate more than 4 cheeseburgers compared to all students.
To find this proportion, we first count the number of data points greater than 4, which are 5 and 7 in this context. We find that there are 2 such data points.
The total number of students, our entire dataset, is 10.

• Proportion calculation is simple—just divide the part by the whole: \(\frac{2}{10} = 0.2\).
• This tells us that 20% of students ate more than 4 cheeseburgers.
• Knowing proportions can be useful for making comparisons or understanding percentages in everyday life.

This process is particularly handy in making sense of data distributions and reporting findings.