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LSD and Tukey yardsticks on three treatments. Also one-way Anova test on same set of data.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The following data arose in a comparison of the effects of hydration on the time taken (in minutes) to complete a set of exercises. There were three groups of subjects; group 1 (fully hydrated), group 2 (partially hydrated) and group 3 (dehydrated).

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Group 1 (fully hydrated)$\\var{r1[0]}$$\\var{r1[1]}$$\\var{r1[2]}$$\\var{r1[3]}$$\\var{r1[4]}$$\\var{r1[5]}$
Group 2 (partially hydrated)$\\var{r2[0]}$$\\var{r2[1]}$$\\var{r2[2]}$$\\var{r2[3]}$$\\var{r2[4]}$$\\var{r2[5]}$
Group 3 (dehydrated)$\\var{r3[0]}$$\\var{r3[1]}$$\\var{r3[2]}$$\\var{r3[3]}$$\\var{r3[4]}$$\\var{r3[5]}$
\n

 

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Using the Yardsticks

\n

The mean values for each group are:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overline{x}_i$
Group 1 $\\var{m1}$
Group 2$\\var{m2}$
Group 3$\\var{m3}$  
\n

The differences between the mean values for the groups are:

\n

Between $1$ and $2=\\;|\\var{m1}-\\var{m2}|=\\var{abs(m1-m2)}$

\n

Between $2$ and $3=\\;|\\var{m2}-\\var{m3}|=\\var{abs(m2-m3)}$

\n

Between $1$ and $3=\\;|\\var{m1}-\\var{m3}|=\\var{abs(m1-m3)}$

\n

We compare these differences with the LSD and Tukey yardsticks:

\n

LSD yardstick = $2.131\\times\\var{sqrms}\\times\\sqrt{2/\\var{n1}}=\\var{lsd}$ to 2 decimal places, where $\\var{sqrms}$ is the value of $\\sqrt{RMS}$ found above.

\n

Tukey yardstick = $3.67\\times\\var{sqrms}\\times\\sqrt{1/\\var{n1}}=\\var{tukey}$ to 2 decimal places.

\n

If the difference of the means:

\n\n\n\n

 

\n

Hence we have the following for the groups:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Pairs of GroupsDefinite Significant DifferencePossible Significant DifferenceNo Significant Difference
Means of Groups 1 and 2{yn[0][0]}{yn[0][1]}{yn[0][2]}
Means of Groups 2 and 3{yn[1][0]}{yn[1][1]}{yn[1][2]}
Means of Groups 1 and 3{yn[2][0]}{yn[2][1]}{yn[2][2]}
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You are given the following ANOVA table for this data:

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
SourcedfSSMSVR
Between Treatments$\\var{dfbt}$$\\var{btss}$$\\var{mbt}$$\\var{vr}$
Residual$\\var{dfrs}$$\\var{rss}$$\\var{mrs}$-
Total$\\var{n-1}$$\\var{tss}$--
\n

 

\n

 Input $\\sqrt{RMS}$ here: [[0]] to 2 decimal places.

\n

 

\n

This will be used to calculate the LSD and Tukey yardstick values later.

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Using ANOVA

\n

Using the $VR$ value given in the table and one-way ANOVA, what is the strength of evidence against the null hypothesis that the mean times taken are the same for the three groups?

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Hence what is your decision based on the above ANOVA analysis?

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Using the Yardsticks

\n

Fill in this table with the appropriate values for the mean values of the groups, all decimals to 2 decimal places:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overline{x}_i$
Group 1[[0]]
Group 2[[1]]
Group 3[[2]]
\n

 

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Now find the LSD and Tukey yardsticks from the above data. Use the value to 2 decimal places you found for $\\sqrt{RMS}$:

\n

   LSD= [[0]]

\n

Tukey= [[1]]

\n

 Using these yardsticks fill in the following table indicating if there is a possible or definite significant difference between the pairs of groups mean times in undertaking the tasks:

\n

[[2]]

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