3.9: Try It!
- Page ID
- 33442
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The weight gain due to 4 different diets given to 24 calves is shown below.
| diet1 | diet2 | diet3 | diet4 |
|---|---|---|---|
| 12 | 18 | 10 | 19 |
| 10 | 19 | 12 | 20 |
| 13 | 18 | 13 | 18 |
| 11 | 18 | 16 | 19 |
| 12 | 19 | 14 | 18 |
| 09 | 19 | 13 | 19 |
a) Write the appropriate null and alternative hypotheses to test if the weight gain differs significantly among the 4 diets.
- Solution
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\(H_{0}: \ \mu_{1} = \mu_{2} = \mu_{3} = \mu_{4}\) vs. \(H_{a}: \ \mu_{i} \neq \mu_{j} for some \(i, j = 1,2,3,4\) OR "Not all means are equal"
Note: Here, \(\mu_{i}, \ i=1,2,3,4\) are the actual mean weight gains due to diet1, diet2, diet3, and diet4, respectively.
b) Analyze the data and write your conclusion.
- Solution
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Using SAS...
data Lesson3_ex1; input diet $ wt_gain; datalines; diet1 12 diet1 10 diet1 13 diet1 11 diet1 12 diet1 09 diet2 18 diet2 19 diet2 18 diet2 18 diet2 19 diet2 19 diet3 10 diet3 12 diet3 13 diet3 16 diet3 14 diet3 13 diet4 19 diet4 20 diet4 18 diet4 19 diet4 18 diet4 19 ; ods graphics on; proc mixed data= Lesson3_ex1 plots=all; class diet; model wt_gain = diet; contrast 'Compare diet1 with diets 2,3,4 combined ' diet 3 -1 -1 -1; store result1; title 'ANOVA of Weight Gain Data'; run; ods html style=statistical sge=on; proc plm restore=result1; lsmeans diet/ adjust=tukey plot=meanplot cl lines; run;
The ANOVA results shown below indicate that the diet effect is significant with an \(F\)-value of 51.27 (\(p\)-value <.0001). This means that not all diets provide the same mean weight gain. The diffogram below indicates the significant different pairs of diets identified by solid blue lines. The estimated mean weight gains from diets 1, 3, 2, and 4 are 11, 13, 18.1, and 19 units respectively. The diet pairs that have significantly different mean weight gains are (1,2), (1,4), (3,2), and (3,4).
Partial Output:
Type 3 Tests of Fixed Effects Effect Num DF Den DF F Value Pr > F diet 3 20 51.27 <.0001 diet Least Squares Means diet Estimate Standard Error DF t Value Pr > |t| Alpha Lower Upper diet1 11.1667 0.5413 20 20.63 <.0001 0.05 10.0374 12.2959 diet2 18.5000 0.5413 20 34.17 <.0001 0.05 17.3708 19.6292 diet3 13.0000 0.5413 20 24.01 <.0001 0.05 11.8708 14.1292 diet4 18.8333 0.5413 20 34.79 <.0001 0.05 17.7041 19.9626 Differences of diet Least Squares Means
Adjustment for Multiple Comparisons: Tukeydiet _diet Estimate Standard Error DF t Value Pr > |t| Adj P Alpha Lower Upper Adj Lower Adj Upper diet1 diet2 -7.3333 0.7656 20 -9.58 <.0001 <.0001 0.05 -8.9303 -5.7364 -9.4761 -5.1906 diet1 diet3 -1.8333 0.7656 20 -2.39 0.0265 0.1105 0.05 -3.4303 -0.2364 -3.9761 0.3094 diet1 diet4 -7.6667 0.7656 20 -10.01 <.0001 <.0001 0.05 -9.2636 -6.0697 -9.8094 -5.5239 diet2 diet3 5.5000 0.7656 20 7.18 <.0001 <.0001 0.05 3.9030 7.0970 3.3572 7.6428 diet2 diet4 -0.3333 0.7656 20 -0.44 0.6679 0.9716 0.05 -1.9303 1.2636 -2.4761 1.8094 diet3 diet4 -5.8333 0.7656 20 -7.62 <.0001 <.0001 0.05 -7.4303 -4.2364 -7.9761 -3.6906 .png?revision=1)
Figure \(\PageIndex{a1}\): SAS-generated diffogram for weight gain comparisons by diet. .png?revision=1)
Figure \(\PageIndex{a2}\): SAS-generated Tukey grouping of weight gains for diet LS-means.
.png?revision=1)
Above is a diffogram depicting the differences in daily commuter time (in hours) among regions of a metropolitan city. Answer the following.
a) Name the regions included in the study.
- Solution
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SOUT, MIDW, NORT, and WEST
b) How many red or blue lines are to be expected?
- Solution
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4 choose 2 = 6 red or blue lines
c) Which pairs of regions have significantly different average commuter times?
- Solution
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(SOUT and NORT), (SOUT and WEST), (MIDW and NORT), and (MIDW and WEST) have significantly different mean commuter times.
d) Write down the estimated mean daily commuter time for each region.
- Solution
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Region SOUT MIDW NORT WEST Estimated mean commuter time in hours 8.7 10.5 16 16.2