The following table shows the weights and prices of some...

The following 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? f. Find and interpret the coefficient of determination using the original data. $$ \begin{array}{|c|c|} \hline \text { Weight (pounds) } & \text { Price } \\ \hline 12.3 & \$ 17.10 \\ \hline 18.5 & \$ 23.87 \\ \hline 20.1 & \$ 26.73 \\ \hline 16.7 & \$ 19.87 \\ \hline 15.6 & \$ 23.24 \\ \hline 10.2 & \$ 9.08 \end{array} $$

The following 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?
f. Find and interpret the coefficient of determination using the original data.
$$
\begin{array}{|c|c|}
\hline \text { Weight (pounds) } & \text { Price } \\
\hline 12.3 & \$ 17.10 \\
\hline 18.5 & \$ 23.87 \\
\hline 20.1 & \$ 26.73 \\
\hline 16.7 & \$ 19.87 \\
\hline 15.6 & \$ 23.24 \\
\hline 10.2 & \$ 9.08
\end{array}
$$

Key Concepts

Coefficient of Determination

The coefficient of determination, denoted as R², represents the proportion of the variance in the dependent variable that can be explained by the independent variable(s) in the regression model. It provides insight into the model's explanatory power, with higher values indicating a better fit between the model and the data.

Scatterplot

A scatterplot is a graphical representation of two quantitative variables where one variable is plotted on the x-axis and the other on the y-axis. This visual tool helps identify patterns, trends, and the potential strength or direction of a relationship between the variables, making it useful for preliminary analyses before using formal statistical methods.

Correlation Coefficient

The correlation coefficient quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where a positive value indicates a direct relationship, a negative value indicates an inverse relationship, and a value near zero suggests little or no linear association.

Linear Regression and Best-Fit Line

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. The best-fit line, typically expressed as y = mx + b, is determined by minimizing the sum of the squares of the vertical distances between the observed values and the values predicted by the line.

Slope and Intercept Interpretation

The slope in a linear regression equation represents the rate at which the response variable is expected to change for a one-unit change in the predictor variable. The intercept is the predicted value of the response variable when the predictor variable is zero. However, the intercept may not be meaningful if zero is not within the scope of the data or if it lacks practical interpretation in context.

Outlier Analysis

Outlier analysis involves examining data points that deviate significantly from the overall trend in a dataset. Outliers can have a substantial impact on statistical measures such as the correlation coefficient and regression line, potentially leading to misleading results. Identifying and understanding the influence of outliers is crucial in both analysis and interpretation.

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