91Ó°ÊÓ

Two are more numbers of mathematical average called mean.

The T Method for Identifying Significantly Different μi ’s

Find value \({Q_{\alpha ,}}I,{I_{\left( {j - 1} \right)}}\)at the Table A.10. in the appendix of the book for given \(\alpha \).

Compute and list the sample means in increasing order. Calculate

\(w = {Q_{\alpha ,}}I,{I_{\left( {j - 1} \right)}}.\sqrt {\frac{{MSE}}{J}} \)

and underline pairs of the sample means that differ by less than W . The pair of sample which are not underscored by the same line corresponding of population or treatment means that they are significantly different.

From the mentioned table, and\(\alpha = 0.05\)

\(\begin{aligned}{l}{Q_{\alpha ,}}I,{I_{\left( {j - 1} \right)}} = {Q_{0.5,5,15,}}\\ = 4.37\end{aligned}\)

The value of

\(MSE = 278.8\)

Compute the w value as

\(w = {Q_{\alpha ,}}I,{I_{\left( {j - 1} \right)}}.\sqrt {\frac{{MSE}}{J}} = 4.37..\sqrt {\frac{{272.8}}{4}} = 36.09\)

First order the sample means

\({\overline x _{3.}} < {\overline x _{1.}}\,\,\, < {\overline x _{4.}}\,\, < {\overline x _{2.}} < {\overline x _{5.}}\)

The following table

The bold values are smaller than w

Start with mean third sample

\({\overline x _{1.}} - {\overline x _{3.}} = 462 - 427.5 = 34.5 < w = 36.09,\)

Indicated that pair \(\left( {3,1} \right)\)underline as pair:

\(\begin{aligned}{l}\,\,\,\,{\overline x _{3.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{1.}}\\\underline {\,\,\,427.5\,\,\,\,\,\,\,462\,\,\,\,\,\,\,\,} \,\end{aligned}\)

Do the same for all pairs

\({\overline x _{4.}} - {\overline x _{3.}} = \,469.3 - 427.5 = 41.8 > w = 36.09\)

So the pair \(\left( {4,3} \right)\) underline as a pair:

\({\overline x _{4.}} - {\overline x _{1.}} = \,469.3 - 462 = 7.3 < w = 36.09\)

SO The pair \(\left( {1,4} \right)\) underline together

\(\begin{aligned}{l}\,\,{\overline x _{1.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{4.}}\\\underline {\,\,\,462\,\,\,\,\,\,\,469.3\,\,\,\,\,\,\,\,} \end{aligned}\)

Next pair is \(\left( {1,2} \right)\)

\({\overline x _{2.}} - {\overline x _{1.}} = 502.8 - 469.3 = 33.5 < w = 36.09\)

The pair\(\left( {1,2} \right)\) should not be underline together.

\({\overline x _{2.}} - {\overline x _{4.}} = 502.8 - 469.3 = 33.5 < w = 36.09\)

This pair \(\left( {4,2} \right)\) should underline together.

\(\begin{aligned}{l}\,\,{\overline x _{2.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{4.}}\\\underline {\,\,\,502.8\,\,\,\,\,\,\,469.3\,\,\,\,\,\,\,\,} \end{aligned}\)

The next pair is\(\left( {5,4} \right)\)

\({\overline x _{5.}} - {\overline x _{4.}} = 532.8 - 469.3 = 62.8 < w = 36.09\)

So the pair \(\left( {4,5} \right)\)should not be underline together.

\({\overline x _{5.}} - {\overline x _{2.}} = 532.1 - 502.8 = 29.3 < w = 36.09\)

So the pair \(\left( {2,5} \right)\)should underline together.

\(\begin{aligned}{l}\,\,{\overline x _{2.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{5.}}\\\underline {\,\,\,515.8\,\,\,\,\,\,\,532.1\,\,\,\,\,\,\,\,} \end{aligned}\)

The connection can be represented

\(\begin{aligned}{l}\,\,{\overline x _{3.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{1.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{4.}}\,\,\,\,\,\,\,\,\,\,\\\underline {\,\,\,437.5\,\,\,\,\,\,\,462.0\,\,\,\,\,\,\,\,\,\,\,\,469.3\,\,\,\,\,\,\,\,} \end{aligned}\)\(\begin{aligned}{l}\,\,{\overline x _{2.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{5.}}\\\underline {\,\,\,515.8\,\,\,\,\,\,\,532.1\,\,\,\,\,\,\,\,} \end{aligned}\)

The conclusion is that there are no significant differences between pairs the a are which are connected with a line: there is significant difference pair that are not connected.

Hence,

\(\begin{aligned}{l}\,\,{\overline x _{3.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{1.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{4.}}\,\,\,\,\,\,\,\,\,\,\\\underline {\,\,\,437.5\,\,\,\,\,\,\,462.0\,\,\,\,\,\,\,\,\,\,\,\,469.3\,\,\,\,\,\,\,\,} \end{aligned}\)\(\begin{aligned}{l}\,\,{\overline x _{2.}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\overline x _{5.}}\\\underline {\,\,\,515.8\,\,\,\,\,\,\,532.1\,\,\,\,\,\,\,\,} \end{aligned}\)