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Midterm Exam Practice for Statistical Analysis Techniques

School
University Of Georgia**We aren't endorsed by this school
Course
STAT 6510
Subject
Statistics
Date
Dec 12, 2024
Pages
4
Uploaded by ChefStar31826
STAT 6120 Midterm Exam PracticeOctober 10, 2024Instructions: The midterm exam will be closed-notes, closed-book.You are allowed touse yourcalculator, a pen or pencil. Please write your name and answers on the answersheets provided. Please do NOT write your answers on the pages with the questions. Partof this exam tests your ability to figure out the easiest way to compute things, based on theinformation provided and the relationships between various quantities.Table 1: Quantile Table of t DistributionDF0.950.9750.99516.3112.7163.6622.924.309.9232.353.185.8461.942.453.71451.682.012.69471.682.012.69481.682.012.68491.682.012.68501.682.012.68511.682.012.68521.672.012.67In thesavingsexample, we are interested in how savings rate is associated with the agedistribution in the population and percent growth rate of per-capita disposable income in acountry. In the data set, we have the values of the following variables forn= 50 countries.sr: savings rate - personal saving divided by disposable incomepop15: percent population under age of 15ddpi: percent growth rate of per-capita disposable income (dpi) in dollarsWe also generate a dummy variablegrpfrompop15 as below:1
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grp: an indicator for a young population country.grp= 1 ifpop15>35 (i.e., if more than35% of the population in the country is under the age of 15), andgrp= 0 ifpop15≤35.1. We first fit a simple linear regression model forsrusinggrpas the only predictor.yi=β0+βxi+ϵi(1)whereyiandxirespectively represent the values of the savings rate and the youngpopulation indicatorgrpfor thei-th country in the data set, withi= 1,· · ·n, andϵiare independent and identically distributed (iid) random errors with a mean of 0 andan unknown varianceσ2.(a) There aren1= 23 countries having young populations out of then= 50 countriesin the dataset. Write this model in the matrix formy=Xβ+ϵ.(b) We obtained some summary statistics ofsrandgrpas follows:Proportion of the countries with young populations in the data set:px=n1/n=0.46Sample mean of the savings rate: ¯y=n−1∑ni=1yi= 9.671Sample variance of the savings rate:vary= (n−1)−1∑ni=1(yi−¯y)2= 20.074Sample mean of the saving rates among countries with young populations:¯y1=n−11∑grpi=1yi= 7.411.Show that the LS estimates ofβarebβ=(XTX)−1XTy=11−px¯y−px¯y1¯y1−¯y=11.60−4.19(2)(c) Show that the true variance ofbβisV arbβ=σ2n1(1−px)px−px−px1,whereσ2is the unknown variance for the random error term.2. Read theRoutput for the linear regression ofsrongrpbelow and answer the questions.savings.fit=lm(sr~grp, savings)summary(savings.fit)Call:lm(formula = sr ~ grp, data = savings)Page 2
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Residuals:Min1QMedian3QMax-10.3263-2.6334-0.13861.782811.1491Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept)11.59630.768815.083< 2e-16 ***grp1-4.18541.1336-3.692 0.000568 ***---Signif. codes:0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Residual standard error:XXXon 48 degrees of freedomMultiple R-squared:0.2212,Adjusted R-squared:XXXF-statistic: 13.63 on 1 and 48 DF,p-value:XXX(a) Fill in the missingResidual standard errorand interpret it.(b) Fill in the missingAdjusted R-squared.(c) Write down the assumptions needed for the F-test shown in the last line of theoutput, write out the null hypothesis, alternative hypothesis, fill in the missingp-value, and explain the result.3. We further includeddpiin the model. Please read theRoutput and answer the questionssavings.fit2=lm(sr~grp+ddpi, savings)summary(savings.fit2)Call:lm(formula = sr ~ grp + ddpi, data = savings)Residuals:Min1QMedian3QMax-9.0792 -2.8720 -0.00082.3691 10.4864Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept)9.82551.02329.603 1.16e-12 ***grp-4.15511.0777-3.8560.00035 ***ddpi0.46750.18912.4730.01707 *---Signif. codes:0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Page 3
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Residual standard error: 3.798 on 47 degrees of freedomMultiple R-squared:0.3109,Adjusted R-squared:0.2815F-statistic:10.6 on 2 and 47 DF,p-value: 0.0001585anova(savings.fit2, savings.fit)Analysis of Variance TableModel 1: sr ~ grp + ddpiModel 2: sr ~ grpRes.DfRSS Df Sum of SqFPr(>F)147 677.85248 766.06 -1-88.208 6.1161 0.01707 *---Signif. codes:0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1(a) Please interpret the estimates of effects ofgrpandddpiand write out the corre-sponding 95% confidence intervals using the t-table provided on page 1.(b) Read the output of R functionanova, write out the null hypothesis, alternativehypothesis, and your conclusion.(c) Based on the results of F-test, theR2and the adjustedR2, which model ofsavings.fitandsavings.fit2 would you recommend?Page 4
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