The following table gives the number of millionaires (in...

The following table gives the number of millionaires (in thousands) and the population (in hundreds of thousands) for the states in the northeastern region of the United States in 2008 . The numbers of millionaires come from Forbes Magazine in March 2007 . a. Without doing any calculations, predict whether the correlation and slope will be positive or negative. Explain your prediction. b. Make a scatterplot with the population (in hundreds of thousands) on the $x$ -axis and the number of millionaires (in thousands) on the $y$ -axis. Was your prediction correct? c. Find the numerical value for the correlation. d. Find the value of the slope and explain what it means in context. Be careful with the units. e. Explain why interpreting the value for the intercept does not make sense in this situation. $$ \begin{array}{|l|c|r|} \hline \text { State } & \text { Millionaires } & \text { Population } \\ \hline \text { Connecticut } & 86 & 35 \\ \hline \text { Delaware } & 18 & 8 \\ \hline \text { Maine } & 22 & 13 \\ \hline \text { Massachusetts } & 141 & 64 \\ \hline \text { New Hampshire } & 26 & 13 \\ \hline \text { New Jersey } & 207 & 87 \\ \hline \text { New York } & 368 & 193 \\ \hline \text { Pennsylvania } & 228 & 124 \\ \hline \text { Rhode Island } & 20 & 11 \\ \hline \text { Vermont } & 11 & 6 \\ \hline \end{array} $$

The following table gives the number of millionaires (in thousands) and the population (in hundreds of thousands) for the states in the northeastern region of the United States in 2008 . The numbers of millionaires come from Forbes Magazine in March 2007 .
a. Without doing any calculations, predict whether the correlation and slope will be positive or negative. Explain your prediction.
b. Make a scatterplot with the population (in hundreds of thousands) on the $x$ -axis and the number of millionaires (in thousands) on the $y$ -axis. Was your prediction correct?
c. Find the numerical value for the correlation.
d. Find the value of the slope and explain what it means in context. Be careful with the units.
e. Explain why interpreting the value for the intercept does not make sense in this situation.
$$
\begin{array}{|l|c|r|}
\hline \text { State } & \text { Millionaires } & \text { Population } \\
\hline \text { Connecticut } & 86 & 35 \\
\hline \text { Delaware } & 18 & 8 \\
\hline \text { Maine } & 22 & 13 \\
\hline \text { Massachusetts } & 141 & 64 \\
\hline \text { New Hampshire } & 26 & 13 \\
\hline \text { New Jersey } & 207 & 87 \\
\hline \text { New York } & 368 & 193 \\
\hline \text { Pennsylvania } & 228 & 124 \\
\hline \text { Rhode Island } & 20 & 11 \\
\hline \text { Vermont } & 11 & 6 \\
\hline
\end{array}
$$

Key Concepts

Scatterplot

A scatterplot is a graphical representation of the relationship between two quantitative variables. It plots data points on a coordinate system, with one variable on the x-axis and the other on the y-axis, allowing one to visually assess patterns, trends, and potential correlations between the variables.

Correlation

Correlation measures the strength and direction of a linear relationship between two quantitative variables. Its value ranges from -1 to 1, with a positive value indicating that as one variable increases, the other tends to increase, and a negative value indicating that as one increases, the other tends to decrease. It provides insight into the degree to which the variables move together.

Slope of the Regression Line

The slope of the regression line indicates the rate of change in the response variable for each unit change in the predictor variable. It is computed from the data using regression analysis and is interpreted in the context of the units of the variables. A positive slope suggests a positive relationship, while a negative slope suggests a negative relationship.

Intercept in Regression

The intercept in a regression model is the predicted value of the response variable when the predictor variable is zero. However, its interpretation can be impractical or meaningless if a zero value is not within the range of observed data or does not make sense in the real-world context of the variables being analyzed.

Prediction Based on Data Trends

Making predictions based on data trends involves using the observed relationship between variables—often visualized through a scatterplot or summarized by the correlation and regression parameters—to forecast the behavior of one variable based on known values of another. This predictive insight helps in understanding and quantifying real-world associations.

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