2018 Senate ElectionA major theme during the 2018 election in both the House and Senate was the popularity of President Trump (a Republican). The scatter diagram below shows the presidential approval rating (percent of registered voters who approve of the job the president is doing) versus the percentage of votes received by the Democratic senatorial candidate in the 2018 election.
c. The least-squares regression equation, where presidential approval (x) is the predictor and percent Democrat (y) is the outcome, is: ŷ = -0.9147x + 94.9933. Interpret the slope. What does this suggest about Senate candidates from the president's party?

Verified step by step guidance

1. Identify the variables in the regression equation: the predictor variable (x) is the presidential approval rating, and the outcome variable (y) is the percentage of votes received by the Democratic senatorial candidate.
2. Look at the slope of the regression equation, which is -0.9147. The slope represents the change in the predicted percentage of votes for the Democratic candidate for each one-unit increase in presidential approval rating.
3. Interpret the slope: since the slope is negative, it means that as the presidential approval rating increases by 1%, the predicted percentage of votes for the Democratic candidate decreases by approximately 0.9147%.
4. Consider what this implies about Senate candidates from the president's party: a higher presidential approval rating is associated with a lower percentage of votes for the Democratic candidate, suggesting that candidates from the president's party tend to receive more votes when the president is more popular.
5. Summarize the interpretation: the negative slope indicates an inverse relationship between presidential approval and the Democratic vote percentage, implying that the president's popularity positively influences the vote share of Senate candidates from the president's party.

Key Concepts

Least-Squares Regression Equation

This equation models the relationship between a predictor variable (x) and an outcome variable (y) by minimizing the sum of the squared differences between observed and predicted values. It provides a line of best fit that can be used to make predictions and interpret trends.

Slope Interpretation in Regression

The slope represents the average change in the outcome variable for each one-unit increase in the predictor variable. A negative slope indicates an inverse relationship, meaning as the predictor increases, the outcome decreases, which helps understand how variables influence each other.

Scatter Plot and Correlation

A scatter plot visually displays the relationship between two quantitative variables. The pattern of points indicates the direction and strength of the correlation, with a downward trend showing a negative correlation, which supports the interpretation of the regression slope.