In this discussion, you will apply the statistical concepts and techniques covered in this week's reading about multiple regression. You will not be completing work in Jupyter Notebook this week. Instead, you will be interpreting output from your Python scripts for the Module Six discussion. If you did not complete the Module Six discussion, please complete that before working on this assignment.

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:

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:

Define the null and alternative hypothesis in mathematical terms and in words.

Report the level of significance.

Include the test statistic and the P$-$value. $($Hint: F$-$Statistic and Prob $($F$-$Statistic$)$ in the output$).$

Provide your conclusion and interpretation of the test. Should the null hypothesis be rejected? Why or why not?

What is the slope coefficient for the weight variable? Is this coefficient significant at $5\%$ level of significance $($alpha$=0\mathrm{.}05)?($Hint: Check the P$-$value, P is greater than the absolute value of t$,$ for weight in Python output. Recall that this is the individual t$-$test for the beta parameter.$)$ See Step $5$ in the Python script.

What is the slope coefficient for the horsepower variable? Is this coefficient significant at $5\%$ level of significance $($alpha$=0\mathrm{.}05)?($Hint: Check the P$-$value, P is greater than the absolute value of t$,$ for horsepower in Python output. Recall that this is the individual t$-$test for the beta parameter.$)$ See Step $5$ in the Python script.

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?

What is the coefficient of determination of your multiple regression model from Module Six? Provide appropriate interpretation of this statistic.OLS Regression Results

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Dep. Variable: mpg R$-$squared: $0\mathrm{.}844$

Model: OLS Adj. R$-$squared: $0\mathrm{.}832$

Method: Least Squares F$-$statistic: $72\mathrm{.}84$

Date: Mon, $12$ Feb $2024$ Prob $($F$-$statistic$)$: $1\mathrm{.}32$e$-11$

Time: $20$:$24$:$52$ Log$-$Likelihood: $-65\mathrm{.}897$

No$.$ Observations: $30$ AIC: $137\mathrm{.}8$

Df Residuals: $27$ BIC: $142\mathrm{.}0$

Df Model: $2$

Covariance Type: nonrobust

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coef std err t P$>|$t$|[0\mathrm{.}0250\mathrm{.}975]$

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Intercept $36\mathrm{.}09991\mathrm{.}48024\mathrm{.}3850\mathrm{.}00033\mathrm{.}06239\mathrm{.}137$

wt $-3\mathrm{.}60010\mathrm{.}567-6\mathrm{.}3480\mathrm{.}000-4\mathrm{.}764-2\mathrm{.}436$

hp $-0\mathrm{.}03070\mathrm{.}008-3\mathrm{.}8350\mathrm{.}001-0\mathrm{.}047-0\mathrm{.}014$

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Omnibus: $7\mathrm{.}491$ Durbin$-$Watson: $2\mathrm{.}114$

Prob$($Omnibus$)$: $0\mathrm{.}024$ Jarque$-$Bera $($JB$)$: $5\mathrm{.}798$

Skew: $0\mathrm{.}995$ Prob$($JB$)$: $0\mathrm{.}0551$

Kurtosis: $3\mathrm{.}826$ Cond. No$.604.$

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