mirror of
https://asciireactor.com/otho/cs-5821.git
synced 2024-11-22 03:25:06 +00:00
86 lines
3.8 KiB
Plaintext
86 lines
3.8 KiB
Plaintext
|
||
|
||
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. |