91Ó°ÊÓ

The table shows the weights and prices of some turkeys at different supermarkets. a. Make a scatterplot with weight on the \(x\) -axis and cost on the \(y\) -axis. Include the regression line on your scatterplot. b. Find the numerical value for the correlation between weight and price. Explain what the sign of the correlation shows. c. Report the equation of the best-fit straight line, using weight as the predictor \((x)\) and cost as the response \((y)\). d. Report the slope and intercept of the regression line, and explain what they show. If the intercept is not appropriate to report, explain why. e. Add a new point to your data: a 30 -pound turkey that is free. Give the new value for \(r\) and the new regression equation. Explain what the negative correlation implies. What happened? $$ \begin{array}{|c|c|} \hline \text { Weight (pounds) } & \text { Price } \\ \hline 12.3 & \$ 17.10 \\ \hline 18.5 & \$ 23.87 \\ \hline 20.1 & \$ 26.73 \\ \hline 16.7 & \$ 19.87 \\ \hline 15.6 & \$ 23.24 \\ \hline 10.2 & \$ 9.08 \\ \hline \end{array} $$

Step by step solution

01

Making a scatterplot and including the regression line

Begin by plotting the given points on a scatter plot with weight on the x-axis and price on the y-axis. Once the points have been plotted, use a calculator or a statistical software to derive the best fit line or the regression line and superimpose that on the scatter plot.

02

Finding the numerical value for the correlation

Next, calculate the correlation between the weight and the price. This can again be done using a calculator or statistical software. Using the formula \(r = \frac{N\sum xy - (\sum x)(\sum y)}{\sqrt{[(N\sum x^2-(\sum x)^2)][(N\sum y^2-(\sum y)^2)]}}\), where \(N\) is the number of points, \(x\) is the weight, and \(y\) is the price. The sign of the correlation shows the direction of the relationship between the two variables. If positive, it indicates that as the weight increases, so does the price. If negative, as weight increases, the price decreases.

03

Reporting the equation of the best-fit line

The equation of the best-fit line, using the least squares method, is \(y = mx + c\), where \(m\) is the slope, \(c\) is the y-intercept, \(x\) is weight, and \(y\) is price. This equation can be obtained from the statistical software or calculator used earlier.

04

Reporting and explaining the slope and intercept of the regression line

The slope represents the average increase in price for each additional pound, while the intercept represents the expected price when the weight is zero. If the intercept is not logically or practically interpretable (like in this case where a weight of zero might not make sense), it would be mentioned that the intercept is not appropriate to report.

05

Adding a new point and providing the new statistical values

Once the new point (30-pound turkey that is free) has been added, the correlation and regression equation should be recalculated. The new point might lead to a significant drop in the correlation, suggesting a negative correlation between the weight and price, meaning that as the weight increases, the price decreases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient

The correlation coefficient is a statistical measure that indicates the strength and direction of a linear relationship between two variables. In this particular exercise, we are interested in understanding how the weight of turkeys relates to their price. The correlation coefficient is denoted by the symbol \( r \). It can be calculated using the formula: \[r = \frac{N\sum xy - (\sum x)(\sum y)}{\sqrt{[(N\sum x^2-(\sum x)^2)][(N\sum y^2-(\sum y)^2)]}}\] Here, \( N \) is the number of observations, \( x \) is the weight, and \( y \) is the price.

• A positive \( r \) indicates a positive relationship, where an increase in weight is associated with an increase in price.
• A negative \( r \) indicates a negative relationship, meaning that as the weight increases, the price decreases.
• A zero \( r \) suggests no linear relationship.

In our updated dataset, including a 30-pound turkey that is free, the correlation coefficient may turn negative, implying a peculiar and mathemtically interesting inverse relationship.

Regression Line Equation

The regression line equation provides a mathematical representation of the relationship between variables. In this case, we're looking at how turkey weight predicts its price. The equation takes the form: \[ y = mx + c \] Where:

• \( y \) is the price.
• \( m \) is the slope of the line.
• \( x \) is the weight (our predictor).
• \( c \) is the y-intercept.

This equation is known as the best-fit line, as it best represents the trend observed in the data using the least squares method. It helps in predicting the turkey's price when its weight is known. In our exercise, the regression line equation changes when a new data point (a 30-pound free turkey) is added, and this is recalculated to reflect the updated data scenario.

Slope and Intercept

The slope and intercept are crucial components of the regression line equation. The slope, denoted by \( m \), indicates how much the price increases for each additional pound of weight. Specifically, it provides insight into the cost per additional pound, serving as a rate of change. The intercept, represented as \( c \), is the predicted price when the weight is zero. In many real-world situations, particularly in this turkey example, a zero-pound turkey does not hold much practical significance. Therefore, the intercept might not be meaningful or reported in such contexts.

• Understanding the intercept helps in some cases, but it should be assessed carefully for applicability.
• The slope offers valuable information about the pricing trend with changes in weight.

The analysis of slope and intercept can help make informed decisions or predictions based on data trends.

Least Squares Method

The least squares method is a fundamental approach in regression analysis used to determine the best-fit line for a set of data points. This technique minimizes the sum of the squares of the differences ("errors") between observed values and the values predicted by the line. Here's why it's crucial:

• It ensures that the line is as close as possible to all data points, offering the best generalization of the observed patterns.
• Minimizing the squared errors accounts for larger discrepancies while maintaining the overall minimum total of those discrepancies.

In our turkey weight-price example, the least squares method helps derive the regression line equation that best predicts a turkey's price from its weight. Even if outliers like the 30-pound free turkey emerge, the least squares method offers a way to adjust accordingly, reflecting the updated predictions.