91Ó°ÊÓ

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

Step by step solution

01

Predict the correlation and slope

Based on a general observation, one can predict that as the population of a state increases, the number of millionaires would also tend to increase. This is a positive correlation. Consequently, the slope will also be positive because the trend is ascending.

02

Create a scatter plot

Plot the data on a scatter plot with the population on the x-axis and the number of millionaires on the y-axis. Confirm that the points exhibit a positive correlation as predicted. This can be observed as an upward trend from left (lower population) to right (higher population).

03

Calculate the correlation

The correlation can be determined using a statistical software. The result should be a positive value, confirming the positive relationship between the population and number of millionaires.

04

Find the slope

The slope can be calculated by determining the change in the y-variable (millionaires) compared to the change in the x-variable (population). This will provide you with the units of millionaires per unit population. It should be positive, reflecting the direct proportionality between population and number of millionaires.

05

Discuss the interpretation of the intercept

The intercept, which is the y-value when x (population in a state) is zero, doesn't make sense in this context. It would hypothetically represent the number of millionaires in a state with a population of zero, which is not possible in reality.

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Key Concepts

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

Scatter Plot

A scatter plot is a type of graph that is used to display and examine relationships between two numeric variables. Consider it as a map of dots that represent individual data points. If we take the exercise given, a scatter plot can help us visualize the relationship between the population of states in the northeastern United States and their respective number of millionaires.

Each point on the scatter plot corresponds to a state's population on the x-axis (independent variable) and the number of millionaires on the y-axis (dependent variable). With the information provided, we'd place a dot for each state at the coordinates representing its population and millionaires. An important feature to look out for in such a plot is the pattern formed by the points. Do they rise from left to right, indicating a positive correlation, or do they fall, indicating a negative correlation?

A well-constructed scatter plot can quickly show us how these two variables are potentially related. In our case, we would expect to see the points forming an upward trend, which would suggest that as the population increases, so does the number of millionaires. This is an essential aspect for beginners to understand as it sets the foundation for more complex statistical analysis such as determining the correlation coefficient or fitting a regression line.

Slope of a Line

The slope of a line in a graph represents the rate of change of the y-variable with respect to the x-variable. In simpler terms, it tells us how steep a line is and in which direction it goes. For every unit increase in the x-variable, the slope shows how much we can expect the y-variable to increase or decrease.

When looking at our exercise, if we imagine drawing an imaginary line through the middle of the scatter plot points that best represents the trend of the data, we'd find the slope of this line. A positive slope—as we predict we will have in this scenario—indicates a positive relationship, which means that as the state population grows, the number of millionaires increases at a consistent rate.

Specifically, if we calculate the slope, we can tell how many additional millionaires are associated with every hundred-thousand-person increase in population. This concrete number can help contextualize data in a real-world setting, making it much easier for students to understand the concept and see its practical applications. Remember to always keep the units in mind, as they provide context to the slope's value.

Statistical Intercept

The statistical intercept, also known simply as the intercept, is the point where the regression line crosses the y-axis. It is the value of y when x is zero. To put it into perspective, it answers the question: If the x-variable were zero, what would the y-variable be?

In many contexts, the intercept has a logical interpretation. However, in our exercise, interpreting the intercept doesn't make sense. Why? Because it would represent the number of millionaires (y) when the population (x) is zero. Clearly, a state cannot have millionaires if there are no people living in it. Hence, trying to interpret the value of the intercept in this case would be akin to answering a nonsensical question.

It's important for students to recognize situations where the intercept does not have a meaningful interpretation to avoid drawing incorrect conclusions from the data. This understanding is critical when applying regression analysis to real-world scenarios, where common sense should align with mathematical results.