The table shows the number of calories in four meals and the cost of each meal.
Cost of Meal and Number of Calories
| Number of calories in the meal | Cost of the meal |
| :----------------------------- | :--------------- |
| 550 | $12 |
| 1,250 | $11 |
| 780 | $13 |
| 650 | $10 |
Which best describes the strength of the model?
A. a weak positive correlation
B. a strong positive correlation
C. a weak negative correlation
D. a strong negative correlation
Answer (1)
Calculate the average number of calories: x ˉ = 807.5 .
Calculate the average cost: y ˉ = 11.5 .
Calculate the correlation coefficient: r ≈ − 0.129 .
The correlation is weak and negative: a weak negative correlation .
Explanation
Analyze the problem We are given a table with the number of calories and the cost of four meals. We need to determine the strength and direction of the correlation between these two variables. A correlation coefficient will help us quantify this relationship.
Understanding Correlation Coefficient The correlation coefficient, denoted as r , measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. A value close to 1 indicates a strong positive correlation, a value close to -1 indicates a strong negative correlation, and a value close to 0 indicates a weak correlation.
Correlation Coefficient Formula The formula for the correlation coefficient is: r = ∑ i = 1 n ( x i − x ˉ ) 2 ∑ i = 1 n ( y i − y ˉ ) 2 ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) where: x i represents the number of calories in the i -th meal, y i represents the cost of the i -th meal, x ˉ is the average number of calories, and y ˉ is the average cost.
Calculate Averages First, calculate the average number of calories: x ˉ = 4 550 + 1250 + 780 + 650 = 4 3230 = 807.5 Next, calculate the average cost: y ˉ = 4 12 + 11 + 13 + 10 = 4 46 = 11.5
Calculate Correlation Coefficient Now, we can calculate the correlation coefficient r . After performing the calculations (using the provided tool), we find that: r ≈ − 0.129
Interpret the Result Since r ≈ − 0.129 , the correlation is weak and negative. This means there is a slight tendency for the cost of the meal to decrease as the number of calories increases, but the relationship is not strong.
Final Answer Based on the calculated correlation coefficient, the best description of the strength of the model is a weak negative correlation.
Real-World Example In real-world scenarios, understanding correlations can help in various fields. For example, in marketing, one might analyze the correlation between advertising spending and sales revenue. A positive correlation would suggest that increased advertising leads to higher sales. However, it's crucial to remember that correlation doesn't imply causation, and other factors might be influencing the relationship.
Examples
In real-world scenarios, understanding correlations can help in various fields. For example, in marketing, one might analyze the correlation between advertising spending and sales revenue. A positive correlation would suggest that increased advertising leads to higher sales. However, it's crucial to remember that correlation doesn't imply causation, and other factors might be influencing the relationship.
