Question

The popular ice cream franchise Coldstone Creamery posted the nutritional information for its ice cream offerings in three serving sizes - "Like it," "Love it," and "Gotta Have it" - on their website. \({ }^{18}\) A portion of that information for the "Like it" serving size is shown in the table. $$\begin{array}{lcc}\hline \text { Flavor } & \text { Calories } & \text { Total Fat (grams) } \\\\\hline \text { Cake Batter } & 340 & 19 \\\\\text { Cinnamon Bun } & 370 & 21 \\\\\text { French Toast } & 330 & 19 \\\\\text { Mocha } & 320 & 20 \\\\\text { OREO }^{\circ} \text { Crème } & 440 & 31 \\\\\text { Peanut Butter } & 370 & 24 \\\\\text { Strawberry Cheesecake } & 320 & 21\end{array}$$ 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 total fat and calories in Coldstone Creamery ice cream.

Short answer

Answer: We should use correlation analysis to analyze the data, as our goal is to understand the relationship between fat content and calorie content. After calculating the Pearson correlation coefficient (r), we can determine the nature of the relationship between total fat and calories in Coldstone Creamery's "Like it" serving size ice cream flavors.

Step-by-step solution

a. Choosing the appropriate analysis method

Linear regression analysis aims to model the relationship between two variables, typically by fitting a linear equation to the observed data. On the other hand, correlation analysis aims to measure the strength and direction of the relationship between two variables. In this case, we are interested in understanding the relationship between fat content and calorie content, so correlation analysis is more appropriate for this task.

b. Analyzing the relationship between total fat and calories

To analyze the relationship between total fat and calories, we need to calculate the Pearson correlation coefficient (r) for the given data. Here are the steps to compute r: 1. Calculate the mean of both fat content (F) and calorie content (C). 2. Calculate the standard deviation of both fat content (F) and calorie content (C). 3. Calculate deviation scores (difference between the observed data point and the mean) for both F and C. 4. Multiply the deviation scores for each flavor of ice cream, i.e., the product of deviation_F and deviation_C. 5. Sum these products over all flavors. 6. Normalize this sum by dividing it by the product of standard deviations of F and C, multiplied by the total number of flavors (n). Once we calculate r, we can interpret it as follows: - If r is close to 1, there is a strong positive correlation between total fat and calories. - If r is close to -1, there is a strong negative correlation between total fat and calories. - If r is close to 0, there is no correlation between total fat and calories. After performing these calculations for the given data, we can determine the nature of the relationship between total fat and calories in Coldstone Creamery's "Like it" serving size ice cream flavors.

Key concepts

Linear Regression Analysis

Linear regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It is commonly used to predict the value of an outcome based on one or more predictors. In simpler terms, we can think of linear regression as a way to draw a line through data points on a graph in such a way that the line summarizes the data's trend.

For example, if we imagine the calorie content of ice cream as the dependent variable (what we want to predict) and the fat content as the independent variable (the predictor), linear regression analysis would help us answer questions like, 'Will increasing the fat content of an ice cream flavor increase its calories?' The result of the regression will provide us a formula that can be used for predicting calorie content from a given fat content.

However, it's important to note that correlation does not imply causation. Therefore, even if linear regression shows us a predictive relationship between fat and calories, it doesn't mean that an increase in fat directly causes the increase in calories; there could be other factors at play that are not considered in the analysis. Nonetheless, by using linear regression, we can estimate the strength and type of relationship between the two variables.

Pearson Correlation Coefficient

The Pearson correlation coefficient, denoted as (r), is a measure of the linear correlation between two variables, providing information on both the strength and direction of the relationship. It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship at all.

When we calculate (r) for the relationship between the fat content and calorie content of ice cream, we're trying to understand how these two variables move in relation to each other. If (r) is positive and close to 1, it suggests that as fat increases, so do the calories, consistently. If (r) is negative and close to -1, it means that as fat content increases, the calorie content tends to decrease, which might be counterintuitive in the context of ice cream nutrition.

To compute (r), statisticians follow the steps outlined in the step-by-step solution, involving finding means, standard deviations, and calculating deviation scores for the observed data. The result guides us to understand the pattern of relationship between the variables in question—a crucial step before deciding if further analysis, like regression, is needed.

Statistical Relationship

A statistical relationship between two variables indicates how one variable may change when the other variable changes. This relationship can be causal—where one variable directly affects the other—or non-causal, where the variables simply tend to change together without necessarily influencing each other.

In the context of our ice cream nutrition analysis, we're exploring whether there's a statistical relationship between total fat and calories. If we observe a consistent pattern—like flavors with more fat also having more calories—it suggests a relationship exists. However, it's critical to remember that correlation does not equal causation. Just because two variables show a relationship does not mean one is the reason for the other's change. There could be underlying factors, such as ingredients or preparation methods, affecting both variables.

Understanding the nature of this relationship can be incredibly valuable. For businesses, it can inform product development and marketing strategies. For consumers, it can guide healthier choices. And for statisticians, it is a primary task to unravel these intricate patterns within data using approaches like linear regression and calculating the Pearson correlation coefficient.