To determine the strength of the correlation between the number of calories in the meal and the cost of the meal, we need to look at the Pearson correlation coefficient. The Pearson correlation coefficient is a measure of the linear correlation between two sets of data and ranges from -1 to 1.
The steps to determine the strength of the correlation are as follows:
1. Observation of Data Points: We have four pairs of data points:
- 550 calories and [tex]$12
- 1250 calories and $[/tex]11
- 780 calories and [tex]$13
- 650 calories and $[/tex]10
2. Calculation of the Correlation Coefficient: The Pearson correlation coefficient is calculated using the formula:
[tex]\[
r = \frac{\sum{(X_i - \overline{X})(Y_i - \overline{Y})}}{\sqrt{\sum{(X_i - \overline{X})^2} \sum{(Y_i - \overline{Y})^2}}}
\][/tex]
where:
- [tex]\( X_i \)[/tex] and [tex]\( Y_i \)[/tex] are the individual sample points.
- [tex]\( \overline{X} \)[/tex] and [tex]\( \overline{Y} \)[/tex] are the mean values of the data series [tex]\(X\)[/tex] and [tex]\(Y\)[/tex].
3. Interpreting the Value: Once the Pearson correlation coefficient [tex]\( r \)[/tex] is calculated, its value indicates the following:
- If [tex]\( r \)[/tex] is close to 1, there is a strong positive correlation.
- If [tex]\( r \)[/tex] is close to -1, there is a strong negative correlation.
- If [tex]\( r \)[/tex] is around 0, it indicates a weak correlation (either positive or negative depending on the sign).
From the data values and their relationships, the calculated Pearson correlation coefficient is approximately -0.129.
4. Analyzing the Coefficient:
- A Pearson correlation coefficient of -0.129 suggests a very weak relationship between the number of calories in a meal and its cost.
- Since the coefficient is negative, it indicates a negative correlation. However, the value is far from -1, so the correlation is not strong.
Given these interpretations, we conclude that there is a weak negative correlation between the number of calories in the meal and the cost of the meal.
Therefore, the best description of the strength of the model is:
- a weak negative correlation
