2017-01-30 06:45:45 +00:00
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1. Describe the null hypotheses to which the p-values given in Table 3.4
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correspond. Explain what conclusions you can draw based on these
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p-values. Your explanation should be phrased in terms of sales , TV ,
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radio , and newspaper , rather than in terms of the coefficients of the
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linear model.
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3. Suppose we have a data set with five predictors, X 1 = GPA, X 2 = IQ,
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X 3 = Gender (1 for Female and 0 for Male), X 4 = Interaction between
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GPA and IQ, and X 5 = Interaction between GPA and Gender. The
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response is starting salary after graduation (in thousands of dollars).
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2017-01-31 02:27:44 +00:00
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Suppose we use least squares to fit the model, and get β₀ = 50, β₁ =
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20, β₂ = 0.07, β₃ = 35, β₄ = 0.01, β₅ = −10.
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2017-01-30 06:45:45 +00:00
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(a) Which answer is correct, and why?
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i. For a fixed value of IQ and GPA, males earn more on average
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than females.
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ii. For a fixed value of IQ and GPA, females earn more on
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average than males.
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iii. For a fixed value of IQ and GPA, males earn more on average
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than females provided that the GPA is high enough.
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iv. For a fixed value of IQ and GPA, females earn more on
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average than males provided that the GPA is high enough.
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(b) Predict the salary of a female with IQ of 110 and a GPA of 4.0.
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(c) True or false: Since the coefficient for the GPA/IQ interaction
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term is very small, there is very little evidence of an interaction
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effect. Justify your answer.
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4. I collect a set of data (n = 100 observations) containing a single
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predictor and a quantitative response. I then fit a linear regression
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model to the data, as well as a separate cubic regression, i.e. Y =
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2017-01-31 02:27:44 +00:00
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β₀ + β₁ X + β₂ X² + β₃ X³ + .
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2017-01-30 06:45:45 +00:00
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(a) Suppose that the true relationship between X and Y is linear,
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2017-01-31 02:27:44 +00:00
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i.e. Y = β₀ + β₁ X + . Consider the training residual sum of
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2017-01-30 06:45:45 +00:00
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squares (RSS) for the linear regression, and also the training
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RSS for the cubic regression. Would we expect one to be lower
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than the other, would we expect them to be the same, or is there
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not enough information to tell? Justify your answer.
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(b) Answer (a) using test rather than training RSS.
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(c) Suppose that the true relationship between X and Y is not linear,
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but we don’t know how far it is from linear. Consider the training
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RSS for the linear regression, and also the training RSS for the
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cubic regression. Would we expect one to be lower than the
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other, would we expect them to be the same, or is there not
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enough information to tell? Justify your answer.
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(d) Answer (c) using test rather than training RSS.
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