Suppose you want to know whether Senate candidates who spend more money on their campaigns receive more votes. For the 35 Senate races in 2020 you collect spending information for every candidate, as well as the percentage vote they received in the election. You run a model predicting the percentage of the vote received with the amount of money spent (in $100,000s) by each candidate and get the following results.
| Coefficient | Standard Error | T | |
|---|---|---|---|
| Spending (in $100,000) | 0.5 | ___ | 0.4 |
| Constant | 34.5 | 5.3 | ___ |
In the first line, you were asked to find the standard error using the Coefficient and the T Statistic. This is done by dividing the Coefficient by the T-Statistic.
\[\hat{S_{\hat{\beta}}} = \hat{\beta}/t\] In this particular case
\[\hat{\beta} = 0.5\]\[t = 0.4\] This gives us the following equation:
\[\hat{S_{\hat{\beta}}} \; =\; 0.5/0.4\; = 1.25\]
The second problem asks us to calculate the T-Statistic for the Constant using the Beta Coefficient and the Standard Error. This is a fairly easy process because the T Statistic is the Beta Coefficient divided by the Standard Error. Since each Standard Error represents one Standard Deviation in a distribution, we can find out how many Standard Deviations from 0 our Coeffiecient is by dividing the Coefficient by the Standard Error.
\[t = \hat{\beta}/\hat{S_{\hat{\beta}}}\] In this particular case
\[\hat{\beta} = 34.5\]\[\hat{S_{\hat{\beta}}} = 5.3\] This gives us the following equation:
\[t \; =\; 34.5/5.3\; = 6.5\]
In the homework, you were asked to estimate the percentage of votes received if they spent $1,000,000. Here’s step by step.
\[{Estimated \: Percent\: of\: Votes\: Received} =\: 34.5 + (0.5\:*\:Spending\: in\: 100,000s)\]
\[ {Estimated \: Percent\: of\: Votes\: Received} =\: 34.5 + (0.5\:*10)\: = 39.5%\] For our Quiz today, let’s work with a new regression output. For this particular analysis, we want to examine the likelihood that an individual is willing to say that they plan on voting for Trump in 2020. We collect some information and have operationalized it as follows.
| Operationalization | Hint | |
|---|---|---|
| Age (in years over 18) | Years Over 18 | If Respondent is 19 then Age = 1 |
| Gender | Male = 0, Female = 1 | ___ |
| Party ID | Republican = 1 | All other parties are 0 |
| Social Desirability (0-3) | Scale from 0 - 3 | 0 = Doesn’t Care What Others Think 3 = Cares A Lot |
| Coefficient | Standard Error | T | |
|---|---|---|---|
| Age (in years over 18) | 0.25 | 0.1 | ___ |
| Gender | -10.0 | ___ | 2.98 |
| Party ID | ___ | 1.10 | 25.00 |
| Social Desirability (0-3) | -3.00 | ___ | 3.25 |
| Constant | 40.5 | 4.3 | ___ |
We will need one final equation to figure all these out.
\[\hat{\beta} = t * \hat{S_{\hat{\beta}}}\]
To calculate the predicted Vote for Trump, we would use the following equation:
\[VoteforTrump = 40.5 + (-3.0*SocialDesirability) + (x*PartyID) + (-10.0*Gender) + (0.25*Ageover18)\]
For your “Quiz” please calculate the T Statistics for Age and the Constant, the Standard Error for Gender and Social Desirability, and the Coefficient for Party ID.