Question

A political scientist was interested in studying America's voting habits. So, he decided to make a least squares regression equation to predict the percentage of people in a state that would vote for Obama in 2012 based on the percentage of people in a state that voted for Obama in 2008. The least squares equation is y-hat = -4.599 +1.04x. In the state of Florida, 50.09% voted for Obama in 2008; whereas, 50.01% voted for him in 2012. What is the value of the residual? Group of answer choices

Answer 1

The residual value for the state of Florida is calculated by subtracting the predicted percentage of votes for Obama in 2012 from the actual percentage. The actual percentage was 50.01% and the predicted was 47.4946%, leading to a residual of 2.5154%.

Step-by-step explanation:

To find the value of the residual for the state of Florida, we need to use the given least squares regression equation, ý-hat = -4.599 +1.04x, where x represents the percentage of people in a state that voted for Obama in 2008, and y-hat represents the predicted percentage of people that would vote for Obama in 2012.

The equation provided by the political scientist to predict the 2012 voting percentages is y-hat = -4.599 + 1.04x. With the voting data for Florida, where 50.09% voted for Obama in 2008, we can predict the 2012 percentage by substituting x = 50.09 into the least squares regression equation:

y-hat = -4.599 + 1.04(50.09) = -4.599 + 52.0936 = 47.4946%

However, the actual percentage that voted for Obama in Florida in 2012 was 50.01%. To find the residual, we subtract the predicted value from the actual value:

Residual = Actual y - Predicted y-hat = 50.01% - 47.4946% = 2.5154%

The value of the residual is 2.5154%. This indicates that there was a 2.5154% higher actual percentage of voters for Obama than what was predicted by the regression line for the state of Florida in 2012.

Answer 2

The value of residual is 2.5131%.

Step-by-step explanation:

We are given the following in the question:

`\hat{y}= -4.599 +1.04x`

The above is a linear regression equation to predict the percentage of people in a state that would vote for Obama in 2012 based on the percentage of people in a state that voted for Obama in 2008.

Independent Variable:

Percentage of people in a state that voted for Obama in 2008

Dependent Variable:

Percentage of people in a state that would vote for Obama in 2012

In the state of Florida, 50.09% voted for Obama in 2008; whereas, 50.01% voted for him in 2012.

Thus, the actual value is 50.01%.

We predict the value from regression equation by putting x = 50.09

`\hat{y}= -4.599 +1.04(50.09) = 47.4946`

The predicted value for percentage of people in a state that would vote for Obama in 2012 is 47.4969%.

Residual is the difference between the predicted value and the actual value.

`\text{Residual} = \text{Observed - Predicted}\\= y-\hat{y}\\= 50.01\% - 47.4969\%\\= 2.5131\%`

Thus, the value of residual is 2.5131%.