cs-5821/hw2/answers

1. Describe the null hypotheses to which the p-values given in Table
   3.4 correspond. Explain what conclusions you can draw based on
   these p-values. Your explanation should be phrased in terms of
   sales , TV , radio , and newspaper , rather than in terms of the
   coefficients of the linear model.


	P-values that are very small indicate that the model for that
	predictor is likely to account for a significant amount of the
	association between the predictor and the response. If that is
	true, then, we reject the null hypothesis, and conclude that a
	relationship exists between the predictor and the response. The
	p-values computed from the response of sales to marketing budget
	for each marketing paradigm indicate will give us insight into
	which of these predictors have a strong relationship with sales
	of this product.

	TV marketing and radio marketing both have a strong relationship
	to sales, according to their linear regression p-values, but
	newspaper advertising does not appear to be effective, given
	that the linear model does not account for much of the variation
	in sales across that domain. We can conclude that cutting back
	on newspaper advertising will likely have little effect on the
	sales of the product, and that increasing TV and radio
	advertising budgets likely will have an effect. Furthermore, we
	can see that radio advertising spending has a stronger
	relationship with sales, as the best-fit slope is significantly
	more positive than the best fit for TV advertising spending, so
	increasing the radio advertising budget will likely be more
	effective.



3. Suppose we have a data set with five predictors, X₁ = GPA, X₂ =
   IQ, X₃ = Gender (1 for Female and 0 for Male), X₄ = Interaction
   between GPA and IQ, and X₅ = Interaction between GPA and Gender.
   The response is starting salary after graduation (in thousands of
   dollars). Suppose we use least squares to fit the model, and get
   β₀ = 50, β₁ = 20, β₂ = 0.07, β₃ = 35, β₄ = 0.01, β₅ = −10.

	(a) Which answer is correct, and why?
		i. For a fixed value of IQ and GPA, males earn more on
		   average than females.

		ii. For a fixed value of IQ and GPA, females earn more on
		average than males.

		iii. For a fixed value of IQ and GPA, males earn more on
		average than females provided that the GPA is high enough.

		iv. For a fixed value of IQ and GPA, females earn more on
		average than males provided that the GPA is high enough.

	(b) Predict the salary of a female with IQ of 110 and a GPA of
	4.0.

	(c) True or false: Since the coefficient for the GPA/IQ
	interaction term is very small, there is very little evidence of
	an interaction effect. Justify your answer.




4. I collect a set of data (n = 100 observations) containing a
   single predictor and a quantitative response. I then fit a linear
   regression model to the data, as well as a separate cubic
   regression, i.e. Y = β₀ + β₁ X + β₂ X² + β₃ X³ + .

	(a) Suppose that the true relationship between X and Y is
	linear, i.e. Y = β₀ + β₁ X + . Consider the training residual
	sum of squares (RSS) for the linear regression, and also the
	training RSS for the cubic regression. Would we expect one to be
	lower than the other, would we expect them to be the same, or is
	there not enough information to tell? Justify your answer.

	(b) Answer (a) using test rather than training RSS.

	(c) Suppose that the true relationship between X and Y is not
	linear, but we don’t know how far it is from linear. Consider
	the training RSS for the linear regression, and also the
	training RSS for the cubic regression. Would we expect one to be
	lower than the other, would we expect them to be the same, or is
	there not enough information to tell? Justify your answer. (d)
	Answer (c) using test rather than training RSS.