Homework 9

Chapter 7

Complete the following 3 problems:

1. Problem 7.6
   • Test the null hypothesis using an \(F\)-test.
   • Did you get a similar \(P\)-value as from the confidence ellipse in Homework 8?.
2. Problem 7.8
   • Follow the same directions as Problem 7.6.
3. Problem 7.9
   • Follow the same directions as Problem 7.6.
   • Hint: Look back at Problem 6.25 from Homework 6.

Chapter 8

Complete the following 3 problems:

1. Problem 8.11
2. Problem 8.15abde
3. Problem 8.16
4. Problem 8.19
5. Problem 8.20

Additional Problems

1. Contrasts for two binary categorical variables without an interaction:
   1. For the model \[ Y = \beta_{0} + \beta_{1} B + \beta_{2} C + \epsilon\] where \(B\) and \(C\) are binary indicator variables for two 2-category categorical variables, solve for the \(\beta\)s in the system of equations \[ \begin{array}{l}{E[Y_{B = 0, C = 0}]=\beta_{0}} \\ {E[Y_{B = 1, C = 0}]=\beta_{0}+\beta_{1}} \\ {E[Y_{B = 0, C = 1}]=\beta_{0}+\beta_{2}} \\ {E[Y_{B = 1, C = 1}]=\beta_{0}+\beta_{1}+\beta_{2}}\end{array}\]
   2. Give an interpretation of the coefficients in terms of contrasts between the categories of the categorical variables.
2. Contrasts for two binary categorical variables with an interaction:
   1. For the model \[ Y = \beta_{0} + \beta_{1} B + \beta_{2} C + \beta_{3} BC + \epsilon\] where \(B\) and \(C\) are binary indicator variables for two 2-category categorical variables, solve for the \(\beta\)s in the system of equations \[ \begin{array}{l}{E[Y_{B = 0, C = 0}]=\beta_{0}} \\ {E[Y_{B = 1, C = 0}]=\beta_{0}+\beta_{1}} \\ {E[Y_{B = 0, C = 1}]=\beta_{0}+\beta_{2}} \\ {E[Y_{B = 1, C = 1}]=\beta_{0}+\beta_{1}+\beta_{2}+\beta_{3}}\end{array}\]
   2. Give an interpretation of the coefficients in terms of contrasts between the categories of the categorical variables.

Extra Credit

Equivalence of the Two-sample \(t\)-test and Regression with a Binary Indicator Variable

Consider the setup for the two-sample \(t\)-test, where we have two independent random samples: \[ \begin{aligned} X_{1}, X_{2}, \ldots, X_{m} &\sim N(\mu_{1}, \sigma^{2}) \\ Y_{1}, Y_{2}, \ldots, Y_{n} &\sim N(\mu_{2}, \sigma^{2}) \end{aligned}\] Note: We are assuming equal variances but allowing unequal sample sizes.

As you learned in introductory statistics, the \(T\)-statistic for a hypothesis test about the difference between the means \(\mu_{1} - \mu_{2}\) is \[ T = \frac{\bar{X} - \bar{Y} - (\mu_{1} - \mu_{2})}{s_{\text{pooled}} \sqrt{1/m + 1/n}}\] where \(s_{\text{pooled}}^{2}\) is the pooled variance \[ s_{\text{pooled}}^{2} = \frac{\left(m-1\right) s_{X}^{2}+\left(n-1\right) s_{Y}^{2}}{m + n -2}.\] This \(T\)-statistic follows a \(t\)-distribution with \(m + n - 2\) degrees of freedom.

Consider an alternative way of encoding the data, where we set up the regression \[ W_{i} = \beta_{0} + \beta_{1} B_{i} + \epsilon_{i}, i = 1, 2, \ldots, m + n\] where \(B_{i}\) is a binary variable with \[ B_{i} = \begin{cases} 0 &: \text{Unit \(i\) is in the second sample} \\ 1 &: \text{Unit \(i\) is in the first sample} \end{cases}\] and we assume that \(\epsilon_{i} \stackrel{\text{iid}}{\sim} N(0, \sigma^{2})\). The \(T\)-statistic for \(\beta_{1}\) is \[ T = \frac{b_{1} - \beta_{1}}{s[b_{1}]},\] which will also be \(t\)-distributed with \(m + n - 2\) degrees of freedom under the SLRGN model assumptions.

In this problem, you will show that the \(T\)-statistic for the two-sample \(t\)-test is equivalent to the \(T\)-statistic for the regression.

1. Find the least squares estimates for \(\beta_{0}\) and \(\beta_{1}\) in the stated simple linear regression in terms of the sample statistics \(\bar{W}, \bar{B},\) and \(s_{B}^{2}\).
2. Determine how the least squares estimates for \(\beta_{0}\) and \(\beta_{1}\) in the stated simple linear regression are related to the sample statistics \(\bar{X}, \bar{Y}, s_{X}^{2}, s_{Y}^{2}, m, \) and \(n\).

   Hint: What are you expecting these estimates to be? That should give you a hint about the simplifications you can make.

3. Find the estimate of the standard error \(s[b_{1}]\) for the stated simple linear regression in terms of \(s_{B}^{2}, \widehat{\sigma_{\epsilon}^{2}}, m, \) and \(n\).
4. Determine how the estimate of the standard error \(s[b_{1}]\) for the stated simple linear regression is related to the sample statistics \(s_{\text{pooled}}^{2}, m, \) and \(n\).

   Hint: What are you expecting the denominator to be? That should give you a hint about the simplifications you can make.

5. Show that the \(T\)-statistic \(T = \frac{b_{1} - \beta_{1}}{s[b_{1}]}\) is equivalent to the \(T\)-statistic \(T = \frac{\bar{X} - \bar{Y} - (\mu_{1} - \mu_{2})}{s_{\text{pooled}} \sqrt{1/m + 1/n}}\), and conclude that both methods perform the same hypothesis test for the difference between the population means.