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

Complete the ANOVA table corresponding to this data, entering values to 2 decimal places.

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SourcedfSSMSVR
Between Treatments$\\var{m-1}$$\\var{btss}$$\\var{msbt}$[[3]]
Between Blocks[[0]][[2]]$\\var{msbb}$$\\var{vrbb} Residual\\var{dfr}$$\\var{rss}$$\\var{rs}$-
Total[[1]]$\\var{tss}$--
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$\\sqrt{RMS}=\\var{sqrms}$ to 2 decimal places. This will be used later to calculate the yardsticks.

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Given the value of $VR$ in the table above, find the range for the $p$ value by using the critical values of $F_{3,12}$ (one-sided) below.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $10\\%$ $5\\%$ $1\\%$ $0.1\\%$ $2.61$ $3.49$ $5.95$ $10.8$
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$p$ less than $0.1\\%$

", "

$p$ lies between $0.1\\%$ and $1\\%$

", "

$p$ lies between $1 \\%$ and $5\\%$

", "

$p$ lies between $5 \\%$ and $10\\%$

", "

$p$ is greater than $10\\%$

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Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the treatments offered by the creams?

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

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Using the yardsticks

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The sample means for the creams are

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W: {me[0]}, X: {me[1]}, Y: {me[2]}, Z: {me[3]}

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Using $q_{t,\\nu}(\\alpha) =4.2$, $t_{\\nu}(\\alpha) =2.179$ and the value for $\\sqrt{RMS}$ above, calculate the LSD and Tukey yardstick.

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LSD yardstick value =    [[0]] (to 2 decimal places).

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Tukey yardstick value = [[1]] (to 2 decimal places).

\n

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Using these yardsticks fill in the following table indicating if there is a possible or definite significant difference between the sample means of pairs of creams.

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To test the effectiveness of pain-relief creams on back pain, five volunteers (A-E) suffering with back pain tried each of four creams (W-Z). The degree of pain relief was measured on a 0-50 scale (higher figures indicate higher levels of pain relief). The results are given below with some totals:

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 A B C D E Totals W {r[0][0]} {r[1][0]} {r[2][0]} {r[3][0]} {r[4][0]} {cols[0]} X {r[0][1]} {r[1][1]} {r[2][1]} {r[3][1]} {r[4][1]} {cols[1]} Y {r[0][2]} {r[1][2]} {r[2][2]} {r[3][2]} {r[4][2]} {cols[2]} Z {r[0][3]} {r[1][3]} {r[2][3]} {r[3][3]} {r[4][3]} {cols[3]} Totals {t[0]} {t[1]} {t[2]} {t[3]} {t[4]} {tot}
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• Test the null-hypothesis using two-way ANOVA that the creams are equally effective.
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• Write down the sample mean for each cream together with the LSD and Tukey yardsticks so that you can see if there is any significant difference between the sample means given by these yardsticks..
• \n
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Using the Yardsticks

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The mean values for each cream are:

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
W X $\\overline{x}_i$ $\\var{me[0]}$ $\\var{me[1]}$ $\\var{me[2]}$ $\\var{me[3]}$
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The differences between the mean values for the creams are:

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Between $W$ and $X=\\;|\\var{me[0]}-\\var{me[1]}|=\\var{abs(me[0]-me[1])}$

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Between $W$ and $Y=\\;|\\var{me[0]}-\\var{me[2]}|=\\var{abs(me[0]-me[2])}$

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Between $W$ and $Z=\\;|\\var{me[0]}-\\var{me[3]}|=\\var{abs(me[0]-me[3])}$

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Between $X$ and $Y=\\;|\\var{me[1]}-\\var{me[2]}|=\\var{abs(me[1]-me[2])}$

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Between $X$ and $Z=\\;|\\var{me[1]}-\\var{me[3]}|=\\var{abs(me[1]-me[3])}$

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Between $Y$ and $Z=\\;|\\var{me[2]}-\\var{me[3]}|=\\var{abs(me[2]-me[3])}$

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We compare these differences with the LSD and Tukey yardsticks:

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LSD yardstick = $2.179\\times\\var{sqrms}\\times\\sqrt{2/\\var{n}}=\\var{lsd}$ to 2 decimal places, where $\\var{sqrms}$ is the value of $\\sqrt{RMS}$ found above.

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Tukey yardstick = $4.2\\times\\var{sqrms}\\times\\sqrt{1/\\var{n}}=\\var{tukey}$ to 2 decimal places.

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If the difference of the means:

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• is greater than the Tukey yardstick we say that there is evidence of a definite significant difference.
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• less than the Tukey yardstick but greater than the LSD yardstick we say that there is evidence of a possible significant difference.
• \n
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• is less than the LSD yardstick then we say that there is no  evidence of a significant difference.
• \n
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Hence we have the following for the creams:

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Pairs of creamsDefinite Significant DifferencePossible Significant DifferenceNo Significant Difference
Means of W and X{yn[0][0]}{yn[0][1]}{yn[0][2]}
Means of W and Y{yn[1][0]}{yn[1][1]}{yn[1][2]}
Means of W and Z{yn[2][0]}{yn[2][1]}{yn[2][2]}
Means of X and Y{yn[3][0]}{yn[3][1]}{yn[3][2]}
Means of X and Z{yn[4][0]}{yn[4][1]}{yn[4][2]}
Means of Y and Z{yn[5][0]}{yn[5][1]}{yn[5][2]}
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