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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. $$ \begin{array}{|l|c|r|} \hline \text { State } & \text { Millionaires } & \text { Population } \\ \hline \text { Connecticut } & 86 & 35 \\ \hline \text { Delaware } & 18 & 8 \\ \hline \text { Maine } & 22 & 13 \\ \hline \text { Massachusetts } & 141 & 64 \\ \hline \text { New Hampshire } & 26 & 13 \\ \hline \text { New Jersey } & 207 & 87 \\ \hline \text { New York } & 368 & 193 \\ \hline \text { Pennsylvania } & 228 & 124 \\ \hline \text { Rhode Island } & 20 & 11 \\ \hline \text { Vermont } & 11 & 6 \\ \hline \end{array} $$

Step by step solution

01

Prediction of Correlation and Slope

Generally, a region with a larger population is likely to have more millionaires, due to a greater number of people and potentially larger economy. Therefore, it is expected that the correlation and slope would be positive.

02

Creation of Scatterplot

Create a scatterplot with population on the x-axis and number of millionaires on the y-axis using a statistical program or tool. With a presumed positive relationship, the dots on the scatterplot should generally move upwards as you go from left to right.

03

Find the Numerical Value for the Correlation

The correlation can be calculated with a statistical tool using the dataset. The resulting value should be positive, confirming the prediction.

04

Slope Value and Meaning

The slope in this context represents how much the number of millionaires increases with each increase in population. Calculate the slope using a statistical tool and interpret the value by considering the unit in context.

05

Intercept Interpretation

The intercept is the y-value when x is zero, i.e., the predicted number of millionaires when the population is zero. Since it's unrealistic to have a population of zero, the intercept in this context doesn't make sense and can't be interpreted.

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Key Concepts

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

Scatterplot Interpretation

A scatterplot is a type of graph used to visualize the relationship between two numerical variables. In our exercise, this involves plotting the population of a region on the x-axis and the number of millionaires on the y-axis.
Each point on the scatterplot represents a state, indicating the state's population in relation to its number of millionaires. A positive relationship means that as the population (x-axis) increases, the number of millionaires (y-axis) also tends to increase.
If your scatterplot shows this upward trend from left to right, it suggests a positive correlation. This is typical in demographic data where larger populations often have more millionaires.

Slope in Context

The slope in regression analysis offers profound insights into the relationship between two variables. In this context, the slope tells us how many more millionaires can be expected with a one-unit increase in the population (measured in hundreds of thousands).
For instance, suppose the slope is found to be 2.5. This indicates that for every additional 100,000 people in population, the number of millionaires increases by 2.5 thousand.

• This slope provides an average rate of change or increase of one variable against the other.
• Understanding the units is crucial in applying this slope practically. Misinterpretation can lead to misunderstandings of the relationship's scale and impact.

Intercept Interpretation

The intercept in a regression equation refers to the expected value of the dependent variable when all independent variables are zero. In our problem, this would conceptually mean the number of millionaires when the population is zero.
In real-world contexts like this one, such an interpretation often doesn't make practical sense because having zero population would logically lead to zero millionaires. Therefore, while mathematically computed, the intercept here serves little practical purpose and is primarily a mathematical artifact deriving from calculations.

• The intercept helps in forming the line of best fit mathematically but should be interpreted cautiously in demographic datasets.
• Ignoring unrealistic contexts like zero population is essential in these interpretations.

Population Data Analysis

Population data analysis involves examining population figures and their implications for other variables, such as wealth distribution, economic growth, or geographical patterns.
In our exercise, population data is crucial for understanding the number of millionaires in various states. Analyzing such datasets can provide insights into economic conditions, disparities, and trends across the states.

• Empirical analysis often relies on such datasets to guide policy decisions.
• Trends in this data can signal shifts in population wealth, influencing economic policymaking.

By leveraging regression analysis, we can make predictions and form insights that are invaluable for economists, policymakers, and researchers focused on demographic studies.