Question

Why does voter turnout vary from election to election? For municipal elections in five different cities, information has been gathered on the percent of registered voters who actually voted, unemployment rate, average years of education for the city, and the percentage of all political ads that used "negative campaigning" (personal attacks, negative portrayals of the opponent's record, etc.). For each relationship: a. Draw a scattergram and a freehand regression line. b. Compute the slope \((b)\) and find the \(Y\) intercept (a). (HINT: Remember to compute b before computing a. A computing table such as Table \(14.3\) is highly recommended.) c. State the least-squares regression line and predict the voter turnout for a city in which the unemployment rate was \(12 \%\), a city in which the average years of schooling was 11, and an election in which \(90 \%\) of the ads were negative. d. Compute \(r\) and \(r^{2} .\) (HINT: A computing table such as Table \(14.3\) is highly recommended. If you constructed one for computing \(b\), you already have most of the quantities you will need to solve for \(r\).) e. Assume these cities are a random sample and conduct a test of significance for each relationship. f. Describe the strength and direction of the relationships in a sentence or two. Which (if any) relationships were significant? Which factor had the strongest effect on turnout? $$ \begin{aligned} &\text { TURNOUT AND UMEMPLOYMENT }\\\ &\begin{array}{ccc} \hline \text { City } & \text { Turnout } & \begin{array}{c} \text { Unemployment } \\ \text { Ratel } \end{array} \\ \hline \text { A } & 55 & 5 \\ \text { B } & 60 & 8\\\ \text { C } & 65 & 9\\\ \text { D } & 68 & 9 \\ \text { E } & 70 & 10 \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { TURNOUT AND LEVEL OF EDUCATION }\\\ &\begin{array}{ccc} \hline \text { City } & \text { Turnout } & \begin{array}{c} \text { Average Years } \\ \text { of School } \end{array} \\ \hline \text { A } & 55 & 11.9 \\ \text { B } & 60 & 12.1 \\ \text { C } & 65 & 12.7 \\ \text { D } & 68 & 12.8 \\ \text { E } & 70 & 13.0 \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { TURNOUT AND NEGATIVE CAMPAIGNING }\\\ &\begin{array}{ccc} \hline \text { City } & \text { Turnout } & \begin{array}{c} \text { percent of negative} \\ \text { Ads } \end{array} \\ \hline \text { A } & 55 & 60 \\ \text { B } & 60 &63\\\ \text { C } & 65 & 55\\\ \text { D } & 68 & 53 \\ \text { E } & 70 & 48 \end{array} \end{aligned} $$

Short answer

Analyze scattergrams, compute the slope and y-intercept for regression lines, and calculate r and r². Conduct statistical tests and summarize findings.

Step-by-step solution

- Draw Scattergrams

Plot scattergrams for each relationship: Turnout vs Unemployment, Turnout vs Education, and Turnout vs Negative Campaigning. Use the city data points given for each scattergram.

- Freehand Regression Line

Draw a freehand regression line that best fits the data points for each scattergram.

- Compute the Slope (b)

Use the formula for the slope: \[b = \frac{N\sum xy - \sum x \sum y}{N\sum x^2 - (\sum x)^2}\], where N is the number of data points, x represents the independent variable (e.g., Unemployment rate, Years of Education, Negative Ads), and y represents the dependent variable (Turnout).

- Find the Y-intercept (a)

Use the formula for the y-intercept: \[a = \bar{y} - b\bar{x}\], where \(\bar{x}\) is the mean of the x values and \(\bar{y}\) is the mean of the y values.

- State the Least-Squares Regression Line

Formulate the least-squares regression line equation using: \[Y = a + bX\]. Predict the voter turnout for scenarios given (e.g., Unemployment rate = 12%, Education = 11 years, Negative ads = 90%).

- Compute the Correlation Coefficient (r) and Coefficient of Determination (r²)

Use the formula for the correlation coefficient: \[r = \frac{N\sum xy - \sum x \sum y}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}}\]. Compute \[r^2\] to understand the proportion of the variance in the dependent variable explained by the independent variable.

- Test of Significance

Assume these cities are a random sample. Conduct a significance test (e.g., t-test) for each relationship to determine if the relationship is statistically significant.

- Describe Relationships

Summarize the strength and direction of each relationship in a sentence or two. Identify which relationships are significant and which factor has the strongest effect on turnout.

Key concepts

Regression Analysis

When we want to understand the relationship between two variables, we often use regression analysis. This mathematical method helps us determine how one variable influences another. For example, in our exercise, we want to know how factors like unemployment, education, and negative campaigning affect voter turnout.

We first plot the data points on a scattergram and draw a line that best represents the data. This line is called the regression line. The formula of this line is: \( Y = a + bX \), where:
* \( Y \) is the dependent variable (voter turnout)
* \(X \) is the independent variable (such as unemployment rate)
* \( a \) is the Y-intercept, the point where the line crosses the Y-axis when X is 0
* \( b \) is the slope, showing how much Y changes when X changes by 1 unit

By computing the slope and intercept, we can draw the regression line and make predictions. This helps us understand and quantify the impact of various factors on voter turnout.

Scattergram

A scattergram, or scatter plot, is a visual tool that allows us to see the relationship between two variables at a glance. Each point on the scattergram represents a pair of values from our data set.

For our exercise, we plotted three different scattergrams:

• Turnout vs Unemployment
• Turnout vs Education
• Turnout vs Negative Campaigning

By plotting these points, we can visually assess whether there is a positive, negative, or no correlation between the variables. A positive correlation means that as one variable increases, the other also increases, whereas a negative correlation means that as one variable increases, the other decreases.

For example, if the scattergram of voter turnout vs education years shows that as education years increase, voter turnout also increases, we can say there is a positive relationship. Drawing a freehand regression line through these points helps us better understand and quantify this relationship.

Correlation Coefficient

The correlation coefficient, denoted as \( r \), measures the strength and direction of the relationship between two variables. It ranges from -1 to 1. Here's what different values of \( r \) indicate:

• 1: Perfect positive correlation (as one variable increases, the other increases)
• 0: No correlation (variables are not related)
• -1: Perfect negative correlation (as one variable increases, the other decreases)

The formula to compute the correlation coefficient is: \[ r = \frac{N \sum xy - \sum x \sum y}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}} \]
In our exercise, we calculated \( r \) for each pair of variables to understand how strongly they are related to voter turnout. For instance, if \( r \) is close to 1 for voter turnout vs education, it suggests a strong positive relationship. Calculating \( r^2 \), the coefficient of determination, helps understand the proportion of variation in one variable explained by the other.

Significance Test

Once we have computed our regression line and correlation coefficient, we need to test whether the relationships we found are statistically significant. This means determining if the observed relationships are due to chance or if they reflect a real underlying pattern.

We use a significance test, like the t-test, to perform this analysis. In our case, we assume the cities in our data are a random sample and check if the relationships between voter turnout and factors like unemployment, education, and negative campaigning are significant.

The t-test helps us determine if the slope of our regression line is significantly different from zero. A significant result (typically, a p-value less than 0.05) means there is a meaningful relationship between the variables, while a non-significant result means the relationship might be due to random variation. Summarizing the strength and direction of these relationships provides valuable insights into which factors most strongly influence voter turnout.