The weight gain due to 4 different diets given to 24 calves is shown below.

1. Write the appropriate null and alternative hypotheses to test if the weight gain differs significantly among the 4 diets.  

   Solution

   \(H_0\colon\mu_1=\mu_2=\mu_3=\mu_4\) vs. \(H_a\colon\mu_i\ne\mu_j\) 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.

    

2. Analyze the data and write your conclusion. 

   Solution

   •  

   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 significantly 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 LS-Means

   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: Tukey
   diet	_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

    

   

    

   

Above is a diffogram depicting the differences in daily commuter time (in hours) among regions of a metropolitan city. Answer the following.

1. Name the regions included in the study.

   Solution

   SOUT, MIDW, NORT, and WEST
2. How many red or blue lines are to be expected?

   Solution

   4 choose 2 = 6 red or blue lines
3. Which pairs of regions have significantly different average commuter times?

   Solution

   (SOUT and NORT), (SOUT and WEST), (MIDW and NORT), and (MIDW and WEST) have significantly different mean commuter times.
4. Write down the estimated mean daily commuter time for each region.

   Answer

   Region	SOUT	MIDW	NORT	WEST
   Estimated mean commuter time in hours	8.7	10.5	16	16.2