Riemann surface
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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.

Solution: In Table 3.4, the null hypothesis for $T V$ is that in the presence of radio ads and newspaper ads, TV ads have no effect on sales. Similarly, the null hypothesis for radio is that in the presence of TV and newspaper ads, radio ads have no effect on sales. (And there is a similar null hypothesis for newspaper.) The low p-values of TV and radio suggest that the null hypotheses are false for TV and radio. The high p-value of newspaper suggests that the null hypothesis is true for newspaper.

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. $Y=\beta_0+\beta_1 X+\beta_2 X^2+\beta_3 X^3+\epsilon$
(a) Suppose that the true relationship between $X$ and $Y$ is linear, i.e. $Y=\beta_0+\beta_1 X+\epsilon$. 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.

Solution: I would expect the polynomial regression to have a lower training $R S S$ than the linear regression because it could make a tighter fit against data that matched with a wider irreducible error $\epsilon$.

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