Last week's discussion involved development of a multiple regression model that used miles per gallon as a response variable. Weight and horsepower were predictor variables. You performed an overall F-test to evaluate the significance of your model. This week, you will evaluate the significance of individual predictors. You will use output of Python script from Module Six to perform individual t-tests for each predictor variable. Specifically, you will look at Step 5 of the Python script to answer all questions in the discussion this week.

In your initial post, address the following items:

1. Is at least one of the two variables (weight and horsepower) significant in the model? Run the overall F-test and provide your interpretation at 5% level of significance. See Step 5 in the Python script. Include the following in your analysis:
   1. Define the null and alternative hypothesis in mathematical terms and in words.
   2. Report the level of significance.
   3. Include the test statistic and the P-value. (Hint: F-Statistic and Prob (F-Statistic) in the output).
   4. Provide your conclusion and interpretation of the test. Should the null hypothesis be rejected? Why or why not?
2. What is the slope coefficient for the weight variable? Is this coefficient significant at 5% level of significance (alpha=0.05)? (Hint: Check the P-value,, for weight in Python output. Recall that this is the individual t-test for the beta parameter.) See Step 5 in the Python script.
3. What is the slope coefficient for the horsepower variable? Is this coefficient significant at 5% level of significance (alpha=0.05)? (Hint: Check the P-value,, for horsepower in Python output. Recall that this is the individual t-test for the beta parameter.) See Step 5 in the Python script.
4. What is the purpose of performing individual t-tests after carrying out the overall F-test? What are the differences in the interpretation of the two tests?
5. What is the coefficient of determination of your multiple regression model from Module Six? Provide appropriate interpretation of this statistic.

Step 5: Multiple regression model to predict miles per gallon using weight and horsepower

This block of code produces a multiple regression model with "miles per gallon" as the response variable, and "weight" and "horsepower" as predictor variables. The ols method in statsmodels.formula.api submodule returns all statistics for this multiple regression model.

Click the block of code below and hit the Run button above.

In[6]:

from statsmodels.formula.api import ols# create the multiple regression model with mpg as the response variable; weight and horsepower as predictor variables.model = ols('mpg ~ wt+hp', data=cars_df).fit()print(model.summary())

OLS Regression Results ============================================================================== Dep. Variable: mpg R-squared: 0.831 Model: OLS Adj. R-squared: 0.818 Method: Least Squares F-statistic: 66.22 Date: Mon, 14 Feb 2022 Prob (F-statistic): 3.88e-11 Time: 12:39:41 Log-Likelihood: -69.962 No. Observations: 30 AIC: 145.9 Df Residuals: 27 BIC: 150.1 Df Model: 2 Covariance Type: nonrobust ============================================================================== coef std err t P>|t| [0.025 0.975] ------------------------------------------------------------------------------ Intercept 37.2710 1.625 22.941 0.000 33.938 40.605 wt -3.9074 0.645 -6.059 0.000 -5.231 -2.584 hp -0.0316 0.009 -3.455 0.002 -0.050 -0.013 ============================================================================== Omnibus: 5.647 Durbin-Watson: 2.078 Prob(Omnibus): 0.059 Jarque-Bera (JB): 4.227 Skew: 0.895 Prob(JB): 0.121 Kurtosis: 3.420 Cond. No. 567. ============================================================================== Warnings: [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.