## 11.5 Exercises

(***): Researchers at IUPUI attempt to predict the number of auto accidents in the city depending on temperature. They randomly select 30 days during the year and run a regression to determine if temperature significantly affected the number of accidents. Using the data*Accidents*`accidents`

, I want you to manually re-create the table we have seen in class to calculate the slope and intercept coefficient and then use R to confirm your result. Note that it is best to copy the`accident`

data and convert it into a regular Excel file for the first part of the exercise.- With temperature as the independent variable and accidents as the dependent variable, create four new columns in Excel: (1) \(x_i-\bar{x}\), (2) \(y_i-\bar{y}\), (3) \((x_i-\bar{x})(y_i-\bar{y})\), and (4) \((x_i-\bar{x})^2\). From there, use the OLS equations provided in the slides to calculate slope and intercept.
- Run a simple bivariate regression using the command
`lm()`

in R and report the results. The results from the calculation with Excel and R must match.

(***): Consider the data sets*Ohio Schools II*`ohioincome`

and`ohioscore`

. In the section on hypothesis testing, the school districts were divided by median income into the top 25% and bottom 25%. In this exercise, two linear regression models are fitted to the data.- In a first step, merge the data sets
`ohioincome`

and`ohioscore`

by IRN. - The first regression model is written as follows: \[score = \beta_0 + \beta_1 \cdot medianincome\] Estimate the above equation using R and report the output. Interpret the coefficient \(\beta_1\). Is it statistically significant?
- Do a scatter plot and include the regression line estimated above in the plot. Is the model a good fit for the data. Compare your answer to the one in the previous part which was based on the numerical output.
- Estimate a second model written as:
\[score = \beta_0 + \beta_1 \cdot medianincome + \beta_2 \cdot medianincome^2\]
For this model, make sure to include the squared term by using the function
`I()`

in R. If you do not include it, R simply drops the last term. Report and interpret the output. - Do a scatter plot and include the (nonlinear) regression line estimated above in the plot. Is the model a good fit for the data. Compare your answer to the previous parts.

- In a first step, merge the data sets
(***): Consider the data set*Indy Home Heating*`heating`

which shows the consumption of natural gas and average temperature. Run a regression with \(usage\) as the dependent variable and \(temperature\) as the independent variable. Interpret the coefficients. Is the variable \(usage\) indicative of your total energy consumption over the time period covered in the data set?