STATISTICS 479 Exam II (100 points)

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1 Name STATISTICS 79 Exam II (1 points) 1. A SAS data set was created using the following input statement: Answer parts(a) to (e) below. input State $ City $ Pop199 Income Housing Electric; (a) () Give the name of a SAS procedure (that is not a graphics procedure) that allows you to construct a normal probability plot of Income and add an inset to the plot. (b) () Give the name of a SAS procedure that you may use to obtain a scatterplot matrix of variables Pop199, Income, Housing, Electric using ODS statistical graphics. (c) () Give the name of a SAS procedure that produces one-way or two-way tables of frequency counts and χ tests of independence for category variables Popgrp and Incomgrp. (d) () Give the name of a SAS statistical graphics procedure to produce histograms of median housing costs in separate panels using data for cities in four population groups. (e) () Give the name of a SAS statistical graphics procedure that you may use to plot horizontal bar graphs that show the averages of electricity usage and group them by a classification variable (such as Region).. (a) (3) Suppose thar the proc step below is in your SAS progam: proc format; value ms 1 = Single = Married 3 = Divorced = Widowed run; Suppose that the variable MaritalStatus has values of 1,, 3 or. Write a SAS statement you may use in any proc step that would make SAS print the formatted values of MaritalStatus. (b) (3) To examine whether a Poisson distribution provides a reasonable model for the number of cell clumps per algae species in a lake the following statement is included in a proc freq step: tables Clumps/ nocum testp=( ); Explain as much as possible the details of the statistical test this statement will produce. (c) (3) The following statement is included in a proc sgplot step using the fueldat dataset as input, where LicGrp and IncomGrp are group variables each with 3 categories: hbar LicGrp/response=Fuel stat=mean group=incomgrp; Explain as much as possible the SAS output this statement will produce. 1

2 (d) (3) The following statement is included in a proc univariate step in a class SAS example using the biology data set as input where Height is a quantitative variable. histogram Height/midpoints=6 to 78 by 3 normal; Explain as much as possible the SAS output this statement will produce. 3. (a) () What is the plot used for comparing a sample distribution to normal distribution using percentiles of each distribution? (b) () The strength of the linear relationship between two variables in a bivariate sample is indicated by the sample correlation coefficient. What plot should you use to verify this relationship? (c) () Name three graphical methods (plots) available for studying the distribution of a (univariate) random sample. (d) () Name a plot that can be used to examine relationships among the variables in multiple regression. (e) () Name a plot to show the values of a quantitative variable (say, population or area) at different values of a category variable (say, state or country).. (a) (3) Examine the proc step that uses the fuel use dataset used in class: proc univariate data=mylib.fueldat cibasic normal mu=; var Income; title "Use of Proc Univariate to Examine Distributions"; run; Describe three statistical results produced by the proc univariate step above for the variable Income: (b) (3) Examine the proc step that uses the fuel use dataset used in class. Suppose Taxgrp variable has two fuel tax levels:low and High. proc ttest data=mylib.fueldat ; class Taxgrp; var Fuel; title "Analyzing Fuel use by Tax groups"; run; Explain as much as possible the important statistical test the above proc ttest step will produce.

3 5. The graph below of side-by-side boxplots summarize life expectancies of 6 countries according to the category variable level of technology. (a) (3) Compare the shapes of the distributions for the four categories of level of technology. Estimate the largest and the smallest life expectancy for all countries (give approx. numbers). (b) (3) Compare the location (as measured by the mean, median) of the distributions for the four categories of level of technology. What trend do you observe in this feature of the disrbutions? (c) (3) Compare the spread or dispersion (as measured by the IQR or range) of the distributions for the four categories of level of technology. Which category of countries (with respect to level of technology) have countries that cover a largest range of life expectancies? (d) (3) Name 3 features in the boxplots you could use to check the model assumptions you made when analyzing this data using an ANOVA model. 3

4 6. Consider following data: Id# x y Id# (contd.) x(contd.) y(contd.) The procedure reg in SAS was used to an perform analysis of this set of data using the model y = β + β 1 x + ɛ where ɛ i are assumed to be independently distributed as N(, σ ) variables. Answer the following questions based on the results appearing in the output attached to the end of the question. i) () What is the value of R? What does this say generally about the fit of this model? ii) () Give the residual sum of squares and its degrees of freedom. What is the estimate s of σ? iii) () What is the value of the F statistic for testing H : β 1 = vs. H a : β 1? Make a decision based on the corresponding p-value and α =.5. iv) () Using the estimate of β 1 and its standard error, compute the t-statistic for testing H : β 1 = vs. H a : β 1. v) () Using the estimate of β 1 and its standard error, compute a 95% confidence interval for β 1. vi) () Use the value of h 33 to compute the standard error of the residual for observation 3. vii) () Use the value of h 33 to compute the standard error of ŷ 3

5 viii) () Use the the residual for observation 3 and its standard error to calculate the corresponding studentized residual. ix) () Use an appropriate plot to determine possible y-outliers. Identify the plot (using the label), give the Id # of the suspected y-outliers. Use RStudent to conduct the Bonferroni test procedure to test if these are y-outliers using Table B.1 supplied using α =.5. x) () Find any cases (use the Id #), if any, that may be x-outliers using a suitable cut-off value. Identify the plot (using the label) that you used to determine if x-outliers are present. Does this plot indicate any x-outliers? xi) () Find any cases from the output statistics (give the Id #), if any, that may be influential explaining why you selected these. Use a plot to determine an influential case, identify this plot (using the label) and say why you selected this case. xii) () If you find any case to be influential, do other related case statistics and/or plots indicate whether this case should be examined carefully. Explain why or why not? xiii) () Identify the residuals vs. predicted values plot (use the label). State what you think this plot indicates about the fit of the data to a straight line model. According to this plot, is the straight line model appear adequate? xiv) () Identify the normal probablity plot of the residuals (use the label). What assumption about the model my be checked using this plot? Explain whether the plot provides evidence to support this assumption. 5

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7 Simple Linear Regression of Data for Exam II: Fall 15 Source Analysis of Variance DF Sum of Squares Mean Square F Value Pr > F Model <.1 Error Corrected Total Root MSE R-Square.678 Dependent Mean 1.5 Adj R-Sq.665 Coeff Var Variable DF Parameter Estimates Parameter Estimate Standard Error t Value Pr > t Intercept x <.1 Obs Id Dependent Variable Predicted Value Std Error Mean Predict Residual Std Error Residual Student Residual Cook's Hat Diag D RStudent H

8 Simple Linear Regression of Data for Exam II: Fall 15 Fit Diagnostics for y Plot A 1 Plot B 1 Plot C Predicted Value Predicted Value Leverage Plot D Plot E Quantile Predicted Value Observation Residual - - Fit Mean Residual Proportion Less Observations Parameters Error DF MSE R-Square Adj R-Square

9 A Formula Sheet A t-test and Confidence Intervals for β m : A t-statistic for testing the hypothesis H : β m = vs. H a : β m is given by t = ˆβ m /{s.e(ˆβ m )} and a (1 α) 1% confidence interval for β m is given by ˆβ m ±t α/,(n k 1) {s.e(ˆβ m )} where t α/,(n k 1) is the upper α/% point of the t-distribution with (n-k-1) d.f., and m =,...,k. Predicted or fitted values: ŷ i = ˆβ + ˆβ 1 x 1i + + ˆβ k x ki, i = 1,...,n Residuals: e i = y i ŷ i, i = 1,...,n Hat Matrix: H = X(X X) 1 X = {h ij } If the diagonal elements of H are h ii for i = 1,...,n and s = mean square error (MSE): Studentized Residuals: Standard Error of ŷ i = = s h ii Standard Error of e i = s (1 h ii ) r i = e i /(s 1 h ii ) for i = 1,...,n. Externally Studentized Residuals: (RStudent) t i = e i /(s (i) 1 hii ) for i = 1,...,n. wheres (i), istheerrormeansquareobtainedfromaregressionmodelfittedwiththeith casedeleted. Leverage: (Hats) Hat or leverage of the i th case is h ii : the diagonal elements of the hat matrix H for i = 1,...,n. Influence Statistics: (Cook s D) D i = 1 k { = 1 k r i } e ( ) i hii s 1 h ii ) ( hii 1 h ii 1 h ii

10 B Tables 539 Table B.1. 5% critical values based on the Bonferroni bounds for the t-test for a single outlier using externally studentized residual in a linear regression model. k n (continued)