$4.$ Simple Linear Regression: Predicting the Total Number of Wins using Average Relative Skill

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You created a simple linear regression model for the total number of wins in a regular season using the average relative skill as the predictor variable.

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See Step $3$ in the Python script to address the following items:

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$$ In general, how is a simple linear regression model used to predict the response variable using the predictor variable?

$$ What is the equation for your model?

$$ What are the results of the overall F$-$test? Use $5\%$ level of significance. Summarize all important steps of this hypothesis test. This includes:

a$.$ Null Hypothesis $($statistical notation and its description in words$)$

b$.$ Alternative Hypothesis $($statistical notation and its description in words$)$

c$.$ Level of Significance

d$.$ Report the test statistic and the P$-$value in a formatted table as shown below:

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Table $1$: Hypothesis Test for the Overall F$-$Test

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Statistic

Value

Test Statistic

X$.$XX

$*$Round off to $2$ decimal places.

P$-$value

X$.$XXXX

$*$Round off to $4$ decimal places.

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e$.$ Conclusion of the hypothesis test and its interpretation based on the P$-$value

$$ Based on the results of the overall F$-$test, can average relative skill predict the total number of wins in the regular season?

$$ What is the predicted total number of wins in a regular season for a team that has an average relative skill of $1550?$ Round your answer down to the nearest integer.

$$ What is the predicted number of wins in a regular season for a team that has an average relative skill of $1450?$ Round your answer down to the nearest integer.

OLS Regression Results $==============================================================================$ Dep. Variable: total$\_$wins R$-$squared: $0\mathrm{.}823$ Model: OLS Adj. R$-$squared: $0\mathrm{.}823$ Method: Least Squares F$-$statistic: $2865.$ Date: Tue, $10$ Dec $2024$ Prob $($F$-$statistic$)$: $8\mathrm{.}06$e$-234$ Time: $13$:$46$:$56$ Log$-$Likelihood: $-1930\mathrm{.}3$ No$.$ Observations: $618$ AIC: $3865.$ Df Residuals: $616$ BIC: $3873.$ Df Model: $1$ Covariance Type: nonrobust $==============================================================================$ coef std err t P$>|$t$|[0\mathrm{.}0250\mathrm{.}975]------------------------------------------------------------------------------$ Intercept $-128\mathrm{.}24753\mathrm{.}149-40\mathrm{.}7310\mathrm{.}000-134\mathrm{.}431-122\mathrm{.}064$ avg$\_$elo$\_$n $0\mathrm{.}11210\mathrm{.}00253\mathrm{.}5230\mathrm{.}0000\mathrm{.}1080\mathrm{.}116==============================================================================$ Omnibus: $152\mathrm{.}822$ Durbin$-$Watson: $1\mathrm{.}098$ Prob$($Omnibus$)$: $0\mathrm{.}000$ Jarque$-$Bera $($JB$)$: $393\mathrm{.}223$ Skew: $-1\mathrm{.}247$ Prob$($JB$)$: $4\mathrm{.}10$e$-86$ Kurtosis: $6\mathrm{.}009$ Cond. No$.2\mathrm{.}14$e$+04==============================================================================$