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1. This problem concerns relating body fat Y (as measured by an expensive, but essentially exact, measure) toa skin measure (Xl) and thigh measure (X2) using linear regression. The Body Fat Example output fits themodel Yi = 130+ 131Xi1 + 132Xi2 + Ci with output from both SAS and R.

(a) What assumptions are being made on the error terms for this analysis to be valid?

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(b) First, state precisely what null hypothesis is being tested by the F statistic of 29.797 in the Analysis ofVariance table (SAS), or equivalently the F-statistic at the end of the summary in the R output?

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(c) What hypothesis is being tested by the P-value of .0369 associated with thigh?

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(d) Notice that the standard errors and the t-statistics for the coefficients are missing. Show with a numberinvolved how with one calculation you can get the standard error for skin from other information on theoutput. Then set-up (with numbers) how you would then get the t-statistic for testing that the coefficientfor skin is O. no need to carry out the calculations.

(e) What is the -0.081628463 in the covariance matrix an estimate of?

eS'j'j·", ,(he J CcJ L 6,/ 'to 2.. J(f) Suppose we fit the model assuming E(Y) = 130+ 131X1 + 132X2 + 133X1X2.

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(g) Suppose we have a third variable X3 = 1 if the subject is white 0 if the subject is non-white. Suppose wefit a model of the form E(Y) = 130+ 131Xi + 132X3 + 133X1 * X3 (note that X2 is not involved here). Giveverbal interpretations to each of the parameters 130, 131, 132 and 133, referring to race as you do so.

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2. A friend of yours took a course which covered an introduction to statistics with just a little regression. Lookingover what you have been doing she sees a number of new things that she doesn't know about. (Let's callher Florence after Florence Nightingale, who provided the motivation for the creation of the Red Cross, andalso happened to be an early member of the American Statistical Association.) Provide a brief answer toFlorence's questions below. Your answers should be: ONE SENTENCE FOR EACH QUESTION ANDSHOULD BE JUST IN WORDS. NO FORMULAS!

(a) I saw this thing labeled consistent covariance of estimates in your SAS output, or what was also referredto as White's robust estimate of the covariance of the estimates in R. Why would I want to use that?I would use that since it estimates THe CG....4C)/} ,/1;J{£" iYAf.li/-}"'ce o·F .b allowing for the fact

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(b) I see that you are doing model building. I remember that R2 was used as a measure of how good themodel was. Why don't you just compute R2 for each combination of variables and chose the combinationof variables with the highest R2?

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(c) I happened to see that example you had modelling yield as a function of temperature and moisture. Sinceit involved temperature- squared and moisture-squared, why do you refer to is as a multiple linear modelwhen the model for expected yield is a non-linear function of yield and moisture, or did you make amistake?I', Mo,! \38 tJCNL:r..vJZ.l)\l ::OJ'l\-1e X'.s BvT\ T l ~l \ w eA ~~' ;!oJ 'TH ~ PAnRl)c If'fl5 (w~ .•cHIS. LV ':AI L"~Iva)f( Ls

fL.\?:F Gnn.:r.:'\.J~'10)(d) What is this Cp statistic used for?

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(e) I know what residuals are. Why do you want to use studentized residuals rather than the regular residuals?

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(f) What is the leverage value used for? Is the leverage value related to the outcome Y?

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Set-up means WITH ALL NUMBERS! BUT NO NEED TO CALCULATE OUT. THE ONLYSELECTION IS FOR TABLE VALUES WHICH YOU CAN LEAVE SYMBOLIC BUT THEREMUST BE NUMBERS FOR THE DEGREES OF FREEDOM INVOLVED.

1. This problem returns to the Fat example introduced in the closed book portion. See the output. Assume theEi are uncorrelated and normally distributed with mean 0 and variance (12.

:r-'" OJ))rv"j a) Set-up how by simply adding two numbers in the output you can find the standard error associated withGflCl()~ getting prediction interval at skin = 19.5 and thigh = 43.1 (which are the values for the first case in the data).

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Sf'll..l $ i'r t.:l a..J,l;""ft+'-< TO b) Suppo~e we wanted simultaneous 90% pre~iction intervals for body fat at 5 different skin/thigh combinations.

'{<JQI10vve- The predictions intervals will be of the form Y ±mult*SEpred. Describe what the multiplier is for the Bonferroniand Scheffe methods and state how you would decide which is better.

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c) Consider estimating p,(XI, X2) = E(YIXI, X2) for a value Xl for skin and X2 for thigh.

- Set-up how you would get an estimate of p,(XI, X2), call this j.l(XI, X2). Your answer should involve Xl and

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- Set-up how you would get an estimate of the standard error of j.l(XI, X2). Your answer can be in a formusing a matrix with numbers and a vector, which could involve numbers and possibly Xl and X2.

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- Using j.l from above and it's estimated standard error, set-up how you would calculate simultaneous 95%confidence intervals for p,(X1, X2) over all values of Xl and X2.

d) There was a third variable,X3' which was a midarm measurement. All you are told is that the SSE for thefit with all three variables is 98.4. Set-up how to carry out a test of size .10 of Ho : (33= 0, where (33 is thecoefficient for X3 in the model with all three variables. Set-up the test statistic (all numbers) and state yourdecision rule in terms of comparing your test statistic to a table value (specified in as much detail as possible).

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phosphorous (other nutrients held fixed). The file has 30 observations including variety (given values of 1and 2), dose and yield, in that order, separated by spaces with no missing values. Consider a model wherefor variety j ( = 1 or 2) with X = dose, E(YIX) = (3jO + (3j1X + (3j2X2. This allows a separate quadraticregression for each variety. Assume the variance is constant throughout.

(b) Provide any additional code needed to run a regression that also has six coefficients but where three ofthem are estimates of (310- (320,(311- (321,and (312- (322. Indicate which coefficients in the output wouldbe estimating these differences. (If you need any additions to the data step beyond what you gave in i)then give them also.)

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(c) Provide any additional code to test the hypothesis that the linear and quadratic terms are the same foreach variety; i.e. the expected value for an observation from variety j is (3jO+ (31X + (32X2.

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3. The second part of the output gives results from workin 'th h P . .has an outcome Y and four predictors. The out ut shows

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variables, etc.) Use this to build a mod 1 . ~. . from all possIble regressIOns (one variable, twofinal model is . Use a p-value of 15 fo : UtSI~gs epwIse s:lectIO~. Explain each step clearly and what the

. r n enng or removmg vanables from the model.

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