Question

The demand for healthy foods that are low in fat and calories has resulted in a large number of "low-fat" or "fat-free" products. The table shows the number of calories and the amount of sodium (in milligrams) per slice for five different brands of fat-free American cheese. $$ \begin{array}{lcc} \text { Brand } & \text { Sodium (mg) } & \text { Calories } \\ \hline \text { Kraft Fat Free Singles } & 300 & 30 \\ \text { Ralphs Fat Free Singles } & 300 & 30 \\ \text { Borden }^{\text {( }} \text { Fat Free } & 320 & 30 \\ \text { Healthy Choice }^{@} \text { Fat Free } & 290 & 30 \\ \text { Smart Beat }^{@} \text { American } & 180 & 25 \end{array} $$ a. Should you use the methods of linear regression analysis or correlation analysis to analyze the data? Explain. b. Analyze the data to determine the nature of the relationship between sodium and calories in fat-free American cheese. Use any statistical tests that are appropriate.

Short answer

Answer: The nature of the relationship between sodium content and calories within fat-free American cheese products across different brands is weakly positive, with a correlation coefficient of approximately 0.3034. This suggests that as sodium content increases, calories tend to increase slightly, but the strength of this relationship is weak, and there can be considerable variability among different brands.

Step-by-step solution

a. Appropriate Method to Analyze the Data

To determine the appropriate method for analyzing the data, let's see what linear regression analysis and correlation analysis provide us: 1. Linear Regression Analysis: this method allows us to predict the value of one variable based on the value of another variable. In this case, we would predict calories based on sodium content. 2. Correlation Analysis: this method determines the strength and direction of the relationship between two variables. In this case, we want to measure the relationship between sodium content and calories within fat-free American cheese. As we are interested in understanding the nature of the relationship between sodium and calories, we should use correlation analysis.

b. Analyzing the Data

Step 1: Calculate the mean of each variable To perform correlation analysis, we first need to determine the mean value for sodium and calories: Mean of Sodium: (300 + 300 + 320 + 290 + 180) / 5 = 1390 / 5 = 278 mg Mean of Calories: (30 + 30 + 30 + 30 + 25) / 5 = 145 / 5 = 29 kcal Step 2: Calculate the covariance Covariance measures the degree to which two variables move together. To calculate the covariance, we need to find each variable's deviation from the mean: $$ COV(X,Y) = \frac{\sum_{i=1}^{n}{(x_i-\bar{X})(y_i-\bar{Y})}}{n} $$ Using the table: $$ \begin{array}{lcccc} \text{Brand} & x_i (x_i-\bar{X}) & y_i (y_i-\bar{Y}) & (x_i-\bar{X})(y_i-\bar{Y})\\ \hline Kraft Fat Free Singles & 300 (22) & 30 (1) & 22\\ Ralphs Fat Free Singles & 300 (22) & 30 (1) & 22\\ Borden Fat Free & 320 (42) & 30 (1) & 42\\ Healthy Choice Fat Free & 290 (12) & 30 (1) & 12\\ Smart Beat American & 180 (-98) & 25 (-4) & 392 \\ \end{array} $$ Therefore, $$ COV(X,Y) = \frac{22 + 22 + 42 + 12 + 392}{5} = \frac{490}{5} = 98 $$ Step 3: Calculate the standard deviations of each variable The standard deviation is the measure of dispersion within a dataset. $$ \sigma_X = \sqrt{\frac{\sum_{i=1}^{n}{(x_i-\bar{X})^2}}{n}} $$ Therefore, $$ \sigma_X = \sqrt{\frac{22^2 + 22^2 + 42^2 + 12^2 + (-98)^2}{5}} $$ $$ \sigma_X = \sqrt{10,468} = 102.31 $$ $$ \sigma_Y = \sqrt{\frac{\sum_{i=1}^{n}{(y_i-\bar{Y})^2}}{n}} $$ Therefore, $$ \sigma_Y = \sqrt{\frac{1^2+1^2+1^2+1^2+(-4)^2}{5}} $$ $$ \sigma_Y =\sqrt{10} = 3.16 $$ Step 4: Calculate the correlation coefficient The correlation coefficient (r) measures the strength and direction of the relationship between two variables: $$ r = \frac{COV(X,Y)}{\sigma_X\sigma_Y} $$ Therefore, $$ r = \frac{98}{102.31 \times 3.16} = \frac{98}{322.68} = 0.3034 $$

Conclusion

We have computed the correlation coefficient (r) to be approximately 0.3034. This value indicates a weak positive relationship between sodium content and calories within these fat-free American cheese products. This suggests that, generally, as the sodium content increases, the calories tend to increase slightly as well; but the relationship is not very strong, and there can be considerable variability among different brands.

Key concepts

Linear Regression Analysis

Linear regression analysis is a statistical method used to understand the relationship between two continuous variables. It helps in predicting the value of one variable based on the value of another. In a linear regression, the goal is to find a linear equation that best predicts the dependent variable from the independent variable. This equation is typically in the form of \( y = mx + c \), where \( m \) is the slope and \( c \) is the intercept.
Although linear regression is powerful, its main use is to predict one variable when the other is known. It can also help identify whether a relationship between two variables is significant, but it's not always the most suited for simply establishing the strength or direction of a relationship without prediction in mind. In the context of sodium and calories in cheese, we are more focused on understanding the association between these variables rather than making an exact prediction.
This is why, for the cheese study, correlation analysis is preferred — it focuses on the relationship itself rather than predicting values.

Covariance

Covariance is an important concept in statistics used to determine how two variables change together. If the two variables increase or decrease together, their covariance is positive. Conversely, if one increases when the other decreases, the covariance is negative. The formula for covariance between two variables \( X \) and \( Y \) is:
\[ COV(X,Y) = \frac{\sum_{i=1}^{n}{(x_i-\bar{X})(y_i-\bar{Y})}}{n} \]
In this equation, \( x_i \) and \( y_i \) are the individual data points for variables \( X \) and \( Y \), while \( \bar{X} \) and \( \bar{Y} \) are their respective means. Covariance can tell us about the direction of the linear relationship between variables, but not its strength or consistency. Thus, it lays foundational ground before computation of the correlation coefficient to better understand variable relationships.
For example, analyzing the sodium and calories in the cheese slices, we get a covariance of 98, indicating that there's a tendency for these variables to increase together. However, it doesn’t specify how strong the trend is, which is where correlation analysis steps in.

Standard Deviation

Standard deviation is a statistical measure that shows the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the values tend to be very close to the mean, while a high standard deviation indicates that the values are spread out over a large range. For a variable \( X \), its standard deviation \( \sigma_X \) can be calculated as:
\[ \sigma_X = \sqrt{\frac{\sum_{i=1}^{n}{(x_i-\bar{X})^2}}{n}} \]
This formula involves calculating the square root of the average squared deviations from the mean \( \bar{X} \). This measure is particularly useful when used alongside other statistical metrics like covariance, as seen here with the cheese data, where standard deviations for sodium and calories were calculated.
By knowing these standard deviations, we can gain insight into the spread of sodium and calorie content across difference brands, which ultimately allows us to calculate the correlation coefficient and understand how consistent and reliable the observed relationships are.

Correlation Coefficient

The correlation coefficient, often represented as \( r \), is a statistical measure that describes the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where:

• -1 indicates a perfect negative linear relationship,
• 0 implies no linear relationship,
• 1 indicates a perfect positive linear relationship.

The formula is:
\[ r = \frac{COV(X,Y)}{\sigma_X\sigma_Y} \]
This equation uses covariance and the product of the standard deviations of the two variables to yield \( r \). In practice, a value close to 1 indicates a strong positive correlation, whereas a value close to -1 suggests a strong negative correlation.
In the cheese example, \( r \) was calculated as approximately 0.3034. This positive but modest correlation suggests that higher sodium tends to come with slightly higher calories, although the relationship isn’t particularly strong. In essence, this metric allows us to both quantify and convey the potential linear association between sodium content and calorie levels in an understandable manner.