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 (2)
Calculate the means of the calories and costs.
Calculate the standard deviations of the calories and costs.
Calculate the covariance between the calories and costs.
Calculate the correlation coefficient: r ≈ − 0.1292 , indicating a weak negative correlation.
The answer is \boxed{a weak negative correlation}.
Explanation
Analyze the problem We have a table of meal costs and calorie counts, and we need to determine the strength and direction of the correlation between them.
Explain the correlation coefficient To determine the correlation, we will calculate the correlation coefficient, denoted as r . The formula for r is: r = σ X ⋅ σ Y Covariance ( X , Y ) where X represents the number of calories, Y represents the cost of the meal, Covariance ( X , Y ) is the covariance between X and Y , and σ X and σ Y are the standard deviations of X and Y respectively.
Calculate the means First, let's calculate the means of the number of calories and the cost of the meals: Mean Calories = 4 550 + 1250 + 780 + 650 = 4 3230 = 807.5 Mean Cost = 4 $12 + $11 + $13 + $10 = 4 $46 = $11.50
Calculate the standard deviations Next, we calculate the standard deviations: Standard Deviation of Calories: σ X = n ∑ ( x i − x ˉ ) 2 = 4 ( 550 − 807.5 ) 2 + ( 1250 − 807.5 ) 2 + ( 780 − 807.5 ) 2 + ( 650 − 807.5 ) 2 σ X = 4 ( − 257.5 ) 2 + ( 442.5 ) 2 + ( − 27.5 ) 2 + ( − 157.5 ) 2 = 4 66306.25 + 195806.25 + 756.25 + 24806.25 σ X = 4 287675 = 71918.75 ≈ 268.1767 Standard Deviation of Costs: σ Y = n ∑ ( y i − y ˉ ) 2 = 4 ( 12 − 11.5 ) 2 + ( 11 − 11.5 ) 2 + ( 13 − 11.5 ) 2 + ( 10 − 11.5 ) 2 σ Y = 4 ( 0.5 ) 2 + ( − 0.5 ) 2 + ( 1.5 ) 2 + ( − 1.5 ) 2 = 4 0.25 + 0.25 + 2.25 + 2.25 = 4 5 = 1.25 ≈ 1.1180
Calculate the covariance Now, we calculate the covariance: Covariance ( X , Y ) = n ∑ ( x i − x ˉ ) ( y i − y ˉ ) Covariance ( X , Y ) = 4 ( 550 − 807.5 ) ( 12 − 11.5 ) + ( 1250 − 807.5 ) ( 11 − 11.5 ) + ( 780 − 807.5 ) ( 13 − 11.5 ) + ( 650 − 807.5 ) ( 10 − 11.5 ) Covariance ( X , Y ) = 4 ( − 257.5 ) ( 0.5 ) + ( 442.5 ) ( − 0.5 ) + ( − 27.5 ) ( 1.5 ) + ( − 157.5 ) ( − 1.5 ) Covariance ( X , Y ) = 4 − 128.75 − 221.25 − 41.25 + 236.25 = 4 − 155 = − 38.75
Calculate the correlation coefficient Finally, we calculate the correlation coefficient: r = σ X ⋅ σ Y Covariance ( X , Y ) = 268.1767 ⋅ 1.1180 − 38.75 = 299.834 − 38.75 ≈ − 0.1292 The correlation coefficient is approximately -0.1292.
Determine the strength and direction of the correlation Since r ≈ − 0.1292 , the correlation is negative because r < 0 . The absolute value of r is ∣ − 0.1292∣ = 0.1292 , which is close to 0. Therefore, the correlation is weak.
Thus, the best description of the strength of the model is a weak negative correlation.
Examples
Understanding the correlation between different variables can be very useful in real life. For example, a store owner might want to know if there is a correlation between the amount spent on advertising and the revenue generated. If the correlation is positive and strong, it means that more advertising leads to more revenue. If the correlation is negative, it means that more advertising leads to less revenue, which could indicate that the advertising strategy needs to be changed. If the correlation is weak, it means that there is little to no relationship between advertising and revenue.
The correlation between meal costs and calories is weakly negative, indicated by a correlation coefficient of approximately -0.1292. This suggests that while there is a slight tendency for lower calorie meals to be more expensive, the relationship is not strong. Therefore, the best choice is option C: a weak negative correlation.
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