Question
A manager at a retail store was interested in the spending habits of customers during the holiday season. The manager surveyed a random group of customers and recorded the number of items that they bought, and the total amount of money they spent at the store. The table displays the information collected on the customers.
Number of Items 1 5 3 8 7 4 3 1 6 8 4 8
Amount Spent (in dollars) 22 92 57 470 204 102 81 85 152 780 52 455
Part A: Would a scatter plot or line graph be more appropriate for displaying and describing this set of data? Explain. (2 points)
Part B: Determine whether the association is linear or nonlinear, strong or weak, and positive or negative and use it to describe the relationship between the number of items purchased and the total amount of money spent
Number of Items 1 5 3 8 7 4 3 1 6 8 4 8
Amount Spent (in dollars) 22 92 57 470 204 102 81 85 152 780 52 455
Part A: Would a scatter plot or line graph be more appropriate for displaying and describing this set of data? Explain. (2 points)
Part B: Determine whether the association is linear or nonlinear, strong or weak, and positive or negative and use it to describe the relationship between the number of items purchased and the total amount of money spent
Answers
GPT-4o mini
### Part A: Choosing Between Scatter Plot and Line Graph
For this data set, a **scatter plot** would be more appropriate than a line graph. This is because scatter plots are used to display the relationship between two quantitative variables—in this case, the number of items purchased (independent variable) and the total amount of money spent (dependent variable). A scatter plot allows us to visualize how the amount spent changes with the number of items bought and helps identify any correlation or trend between the two variables.
In contrast, a line graph is typically used for displaying data trends over time or categorical data, which is not suitable for this data set, as the values do not represent time or ordered categories but individual purchases.
### Part B: Analyzing the Association
To determine the nature of the association, we can look at the relationship between the number of items purchased and the amount spent:
1. **Linear or Nonlinear**: By visualizing the data in a scatter plot, we would examine the pattern formed by the points. If the points roughly follow a straight line, the relationship is linear; if they curve or are spread widely with no clear pattern, it is nonlinear. Given the data, it appears that as the number of items increases, the total amount spent also tends to increase but may have some variations (for example, large amounts spent for fewer items in some cases), which suggests a possible nonlinear relationship.
2. **Strong or Weak**: The strength of the relationship can be assessed by how tightly the data points cluster around a line (if there is one). If there are many points scattered away from a potential line, the strength is weak. In this case, the variations in spending amounts at different quantities of items imply a weaker association.
3. **Positive or Negative**: The relationship appears to be **positive**, as generally, as the number of items purchased increases, the amount spent also increases. Even though there may be some outliers (e.g., a very high amount spent for 8 items), the overall trend remains positive.
### Summary of the Relationship
In summary, the relationship between the number of items purchased and the total amount spent can be described as a nonlinear, weak, and positive association. As customers buy more items, they tend to spend more money; however, there are notable exceptions and variations which lessen the strength of this association.
For this data set, a **scatter plot** would be more appropriate than a line graph. This is because scatter plots are used to display the relationship between two quantitative variables—in this case, the number of items purchased (independent variable) and the total amount of money spent (dependent variable). A scatter plot allows us to visualize how the amount spent changes with the number of items bought and helps identify any correlation or trend between the two variables.
In contrast, a line graph is typically used for displaying data trends over time or categorical data, which is not suitable for this data set, as the values do not represent time or ordered categories but individual purchases.
### Part B: Analyzing the Association
To determine the nature of the association, we can look at the relationship between the number of items purchased and the amount spent:
1. **Linear or Nonlinear**: By visualizing the data in a scatter plot, we would examine the pattern formed by the points. If the points roughly follow a straight line, the relationship is linear; if they curve or are spread widely with no clear pattern, it is nonlinear. Given the data, it appears that as the number of items increases, the total amount spent also tends to increase but may have some variations (for example, large amounts spent for fewer items in some cases), which suggests a possible nonlinear relationship.
2. **Strong or Weak**: The strength of the relationship can be assessed by how tightly the data points cluster around a line (if there is one). If there are many points scattered away from a potential line, the strength is weak. In this case, the variations in spending amounts at different quantities of items imply a weaker association.
3. **Positive or Negative**: The relationship appears to be **positive**, as generally, as the number of items purchased increases, the amount spent also increases. Even though there may be some outliers (e.g., a very high amount spent for 8 items), the overall trend remains positive.
### Summary of the Relationship
In summary, the relationship between the number of items purchased and the total amount spent can be described as a nonlinear, weak, and positive association. As customers buy more items, they tend to spend more money; however, there are notable exceptions and variations which lessen the strength of this association.