Answer in Markdown 2 Points each
What assumption do we make on the noise terms when doing linear regression? How can we check it?
Your friend tells you that it's important to minimize both the SSR and TSS. What's wrong with minimizing the TSS?
How do you justify the presence of a slope?
What is the best numeric value or statistic for justifying the existence of a correlation?
What should you plot to justify an ordinary 4-dimensional least squares regression?
Why do we use different number of deducted degrees of freedom when doing hypothesis testing vs performing the regression?
Write a model equation for 3-dimensional ordinary least squares regression with an intercept. For example, a one dimensional model equation without an intercept would be $y = \beta_0 x + \epsilon$
Write a model equation for when $y \propto \ln{x}$. Assume no intercept
Write a model equation for a person's life expectancy ($l$) assuming it depends on gender ($s$) and if the person eats vegetables ($v$). Assume for this problem that gender and eating vegetables are both binary (0 or 1).
Write a model equation for homework performanced ($h$) based on music genre listended to while working. The following genres are conisdered: Kwaito, Electroswing, and Djent Metal. You can only listen to one genre at a time. Use the letters $k$, $e$, and $d$.
Answer in Python or Markdown as appropiate 4 Points each
If $\sigma_{xy} = -2.1$, $\sigma_{x}^2 = 3.5$, $\sigma_{y}^2 = 1.7$, what is the best fit slope? How does it change if the intercept is $-2.1$?
If your model equation is $y = \beta_0 + \beta_1 x + \beta_2 z + \epsilon$, what is the deducted degrees of freedom?
If $N = 12$, $D = 2$, and $S^2_{\beta_0} = 2.5$, what is the width of a 90% confidence interval for $\beta_0$?
If your best fit intercept is $\hat{\alpha} = 3$ with a standard error of $0.7$, what is the $p$-value for the existence of the that intercept? Take $N = 15$ and assume it's 1D OLS.
Regress the following data to the model equation $y = \beta_0 \ln x + \beta_1 x + \beta_2 +\epsilon$ using a linearization so that you use ND OLS. Report the following:
x = [0.2, 0.29, 0.39, 0.48, 0.57, 0.66, 0.76, 0.85, 0.94, 1.04, 1.13, 1.22, 1.31, 1.41, 1.5]
y = [2.92, 2.58, 3.18, 4.27, 4.5, 3.93, 4.32, 4.57, 4.55, 4.7, 5.02, 4.21, 3.04, 4.98, 6.45]