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Final Exam
Psychology 406
Fall 2007
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Name: \rule{5in}{.02in}
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Whenever possible, use fractions throughout.
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I. Suppose I am given the following set of observations on six individuals. Variable $Y$ refers to the score on a test of reasoning; $X$ is a dichotomous variable indicating the experimental condition to which the subject was randomly assigned (i.e., a value of 1 for one condition; 0 for a second).
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\begin{tabular}{ccc}
Subject & $Y$ & $X$ \\ [2ex] \hline
1 & 7 & 1 \\
2 & 4 & 1 \\
3 & 1 & 1 \\
4 & 2 & 0 \\
5 & 0 & 0 \\
6 & 1 & 0 \\
\end{tabular}
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\noindent
In the questions below, make sure you outline your work.
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a) In the regression of $Y$ on $X$, the model can be represented in the matrix form $\mathbf{Y} = \mathbf{X} \mbox{\boldmath $\beta$} + \mbox{\boldmath $\epsilon$}$. Write this out explicitly by putting in numerical values within $\mathbf{Y}$ and $\mathbf{X}$.
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b) Using \emph{matrix methods}, find the least squares estimate of $\mbox{\boldmath $\beta$}$ (i.e., use the general form of the solution to the normal equations).
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c) Plot the raw score regression equation on a scatterplot of $Y$ versus $X$. Provide a table of predicted values for each of the six observations.
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d) Give the sample standardized regression coefficient corresponding to $\beta_{1}$.
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e) Find the sample correlation between $Y$ and $X$ and relate it to your answer in (d).
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f) Generate the analysis-of-variance table including the sources of variation, the various sum-of-squares and mean squares, degrees of freedom, and F-ratio.
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g) Provide the 95\% confidence interval on $\beta_{0}$ and $\beta_{1}$ (remembering that for a $t$-distribution with 4 degrees of freedom,
P(-2.776 $ < t_{4} < $ -2.776) = .95).
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h) Construct the 95\% confidence intervals on the expected value of $Y$ when $X$ is 0 and when $X$ is 1.
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i) Obtain the 95\% prediction intervals on $Y$ when $X$ is 0 and when $X$ is 1.
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j) Can one obtain a sum-of-squares for pure error in this simple data set? If so, find it; otherwise, explain when you can't find it.
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k) Find the sample means of the $Y$ observations when $X$ is 1 and when $X$ is 0. Show numerically the relationship between $b_{0}$ and $b_{1}$ found in (b) and these two means.
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II. The following data (from William Rohwer) were obtained from 32 students selected (supposedly) at random from an upper-class residential school. He was interested in determining how well data from several paired associate (PA) learning-proficient tests may be used to predict children's performance on the Peabody Picture Vocabulary Test (PPVT). The independent variables were the sum of the number of items correct out of 20 (on two exposures) to 3 types of PA tasks: NS (named still); NA (named action); and SS (sentence still).
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\begin{tabular}{ccccc}
Subject & PPVT & NS & NA & SS \\ [2ex] \hline
1 & 68 & 8 & 21 & 22 \\
2 & 82 & 21 & 28 & 21 \\
3 & 82 & 17 & 31 & 30 \\
4 & 91 & 16 & 27 & 25 \\
5 & 82 & 21 & 28 & 16 \\
6 & 100 & 18 & 32 & 29 \\
7 & 100 & 17 & 26 & 23 \\
8 & 96 & 11 & 22 & 23 \\
9 & 63 & 14 & 24 & 20 \\
10 & 91 & 16 & 27 & 30 \\
11 & 87 & 17 & 25 & 24 \\
12 & 105 & 10 & 26 & 22 \\
13 & 87 & 14 & 25 & 19 \\
14 & 76 & 18 & 27 & 22 \\
15 & 66 & 3 & 16 & 11 \\
16 & 74 & 11 & 12 & 15 \\
17 & 68 & 10 & 28 & 23 \\
18 & 98 & 12 & 30 & 18 \\
19 & 63 & 13 & 19 & 16 \\
20 & 94 & 14 & 27 & 19 \\
21 & 82 & 16 & 21 & 24 \\
22 & 89 & 15 & 23 & 28 \\
23 & 80 & 14 & 25 & 24 \\
24 & 61 & 11 & 16 & 22 \\
25 & 102 & 17 & 26 & 15 \\
26 & 71 & 8 & 16 & 14 \\
27 & 102 & 21 & 27 & 31 \\
28 & 96 & 20 & 28 & 26 \\
29 & 55 & 19 & 20 & 13 \\
30 & 96 & 10 & 23 & 19 \\
31 & 74 & 14 & 25 & 17 \\
32 & 78 & 18 & 27 & 26 \\
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The SYSTAT analyses given on the last attached pages were carried out to obtain the basic statistics, correlations, and regression models involving 0, 1, 2, or 3 of the independent variables. You should use these analyses in answering questions (a) through (h). Make sure your reasoning and work are shown!
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Consider the analysis that includes all three independent variables (NS, NA, SS) in predicting PPVT, and note that P(-2.042 $ < t_{30} < $ 2.042) = .95).
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a) Show numerically using the information given only in the analysis-of-variance table that the squared multiple correlation is .347 and the adjusted squared multiple correlation is .277.
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b) Give a 95\% confidence interval of the (unstandardized) regression coefficient for NA.
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c) Show numerically how the test on the regression coefficient for NA against 0 can be obtained from the ``extra sum of squares'' strategy using the regression analysis involving all three independent variables and the regression analysis involving the two independent variables NS and SS.
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d) The standardized coefficient for NA is .514. Show numerically how this is obtained from the unstandardized coefficient and the basic statistics.
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e) What is the squared multiple correlation in predicting NA from NS and SS?
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f) From the information given in the analysis using three independent variables, what is P(-.478 $ < t_{30} < $ .478)?
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g) Instead of defining the \emph{optimal} linear combination of NS, NA, and SS to predict PPVT, suppose we merely decide to weight all three independent variables equally, i.e., NA + NA + SS. Obtain the correlation between this equally weighted composite and PPVT (hint: we can use our formulas for covariances/variances of linear combinations). Why must the square of this correlation be less than or equal to .34?
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III. Suppose I have the following two vectors and one matrix:
\[ \mathbf{A} = \left[ \begin{array}{r}
1 \\
0 \\
-1\\
\end{array} \right] ; \]
\[ \mathbf{B} = \left[ \begin{array}{l}
2 \\
1 \\
0 \\
\end{array} \right] ; \] \[ \mathbf{C} = \left[ \begin{array}{lll}
3 & 1 & 3 \\
0 & 2 & 2 \\
4 & 1 & 2 \\
\end{array} \right] \]
Find:
a) \[(\mathbf{A}'\mathbf{B})^{-1} = \]
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b)\[ \mathbf{C}\mathbf{B} = \]
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\noindent IV. Completion:
\noindent 1) In a simple regression of $Y$ on $X$, the smaller the absolute value of $r_{xy}$, the \rule{2in}{.01in} the standard error of estimate.
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\noindent 2) In a multiple linear regression, a linear dependency or near linear dependency among the independent variables is called \rule{2in}{.01in}.
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\noindent 3) One of the basic assumptions used in analysis of covariance is that the within-group regression coefficients (of the dependent measure on the covariate) are \rule{2in}{.01in}.
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\noindent 4) In a multiple linear regression of $Y$ on $X_{1}, \ldots, X_{p-1}$, the correlation between predicted (fitted) and obtained values on $Y$ is called the \rule{3in}{.01in}.
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\noindent 5) If $X$ and $Y$ are two random variables, $E\{(X-E(X))(Y-E(Y))\}$ is called the \rule{2in}{.01in}.
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\noindent 6) The expression $(1/2)\log_{e}([1+C]/[1-C])$ is used to define \rule{2in}{.01in} transformation.
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\noindent 7) In generating estimates for a set of regression coefficients for predicting $Y$ from $X_{1}, \ldots, X_{p-1}$, the criterion typically optimized is the sum of the squared deviations of the values on $Y$ from those predicted from the regression equation. This is referred to as the \rule{2in}{.01in} criterion.
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\noindent 8) The square of the multiple correlation coefficient is called the \rule{2in}{.01in}.
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\noindent 9) The Gauss-Markov theorem states that our usual regression estimates have \rule{2in}{.01in} among all unbiased \rule{2in}{.01in} estimates.
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\noindent 10) Assuming the usual normal error model in regression, the least-squares estimates of the regression coefficient are also \rule{2in}{.01in} estimates.
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\noindent 11) In predicting $Y$ from $X$, if certain values on $X$ are repeated, the error sum of squares can be decomposed into two parts: \rule{2in}{.01in} and \rule{2in}{.01in}.
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\noindent 12) Square matrices that have full rank have \rule{2in}{.01in} and therefore are called nonsingular.
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\noindent 13) When we assess how well a sample regression equation performs on a new sample of observations, we are carrying out the task of \rule{2in}{.01in}.
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\noindent 14) One robust strategy for constructing confidence intervals on population variances is based on the sample reuse strategy referred to as the \rule{2in}{.01in}. (two answers are possible here)
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\noindent 15) Given three variables, $Y$, $X_{1}$, and $X_{2}$, if $Y$ is correlated with the discrepancies between $X_{1}$ and $\hat{X_{1}}$ , where $\hat{X_{1}}$ are the fitted values in predicting $X_{1}$ from $X_{2}$, this correlation is called the \rule{2in}{.01in} correlation and is denoted by \rule{2in}{.01in}.
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\noindent 16) In predicting $Y$ from one independent variable $X$, which of the following expressions are zero ($\hat{Y}$ refers to the fitted values):
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a) $\sum (Y_{i} - \hat{Y}_{i})$
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b) $\sum Y_{i}\hat{Y}_{i}$
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c) $\sum X_{i}(Y_{i} - \hat{Y}_{i})$
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d) $\sum Y_{i}(Y_{i} - \hat{Y}_{i})$
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e) $\sum X_{i}\hat{Y}_{i}$
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\noindent 17) The jackknife estimate of a parameter is the \rule{2in}{.01in} of the pseudovalues.
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\noindent 18) In a simple regression model that includes only the additive constant, the sum of squares total and the sum of squares \rule{2in}{.01in} are equal.
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\noindent 19) In a multiple regression of $Y$ on $X_{1}$ and $X_{2}$, $\mathrm{SSR}(X_{1} \mid X_{2}) / \mathrm{SSE}(X_{2})$ is \rule{6in}{.01in}.
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\noindent 20) Given a set of variables $X_{1}, X_{2}, \ldots, X_{p-1}$ obtained on $n$ individuals belonging to two groups, that linear combination of $X_{1}, X_{2}, \ldots, X_{p-1}$ maximizing the square of the usual $t$-statistic is called \rule{4in}{.01in}.
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V. True-False:
\rule{1in}{.01in} 1) In general, if the covariance between two random variables is zero, then $X$ and $Y$ are independent.
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\rule{1in}{.01in} 2) Assuming $N$ independent observations from a bivariate normal distribution, the sample correlation is an unbiased estimate of the population correlation.
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\rule{1in}{.01in} 3) The regression equation found for a sample is equally good as an approximation to the population rule over all the different values of the independent variable.
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\rule{1in}{.01in} 4) In using a linear rule for prediction, it is always a good bet that an individual will fall relatively closer to the group mean on the variable actually known than on the thing predicted.
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\rule{1in}{.01in} 5) If a joint bivariate normal distribution is assumed for two random variables $X$ and $Y$, then inferences about correlation are equivalent to inferences about independence or dependence between the two random variables.
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\rule{1in}{.01in} 6) Assuming bivariate normality, the sampling distribution of the sample correlation coefficient is symmetric around the population value of the correlation coefficient.
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\rule{1in}{.01in} 7) Regression toward the mean depends solely on the use of a \emph{linear} rule for predicting one variable from the other.
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\rule{1in}{.01in} 8) Increasing the variance of an independent variable by multiplying by a constant will increase the standardized regression coefficient by that same multiplicative amount.
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\rule{1in}{.01in} 9) If $X$ and $Y$ are bivariate normal, then the conditional variance of $Y$ given $X = x$ increases as $x$ increases.
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\rule{1in}{.01in} 10) In a multiple regression model involving two independent variables, the test of $H_{o}: \beta_{1} = 0$ and $\beta_{2} = 0$ is equivalent to the separate tests of $H_{o}: \beta_{1} = 0$ and $H_{o}: \beta_{2} = 0$
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