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QUESTION In a horticultural experiment three varieties of tomato
plant are grown.  The number $n$ of plants of each variety and the
yield $x$ (in kg) of each plant are summarized in the table below

$$\begin{array} {rccc} {\rm variety} & n & \sum x & \sum x^2\\
{\rm Money-Maker(M)} & 8 & 95 & 1160\\ {\rm Tigerella}(T)  6 & 92
& 1430\\ {\rm Outdoor\ Girl}(G) & 6 & 76 & 1000 \end{array}$$

Assuming that the yields of each variety are normally distributed
about means $\mu_M,\mu_T,\mu_G$ respectively with common variance
$\sigma^2$.

\begin{description}
\item[(i)]
Estimate $\sigma^2$.

\item[(ii)]
Test the hypothesis $\mu_M=\mu_T=\mu_G$.

\item[(iii)]
Set up a $95\%$ confidence interval for $\mu_M-\mu_T$
\end{description}


ANSWER
 $n=20\ \ T=95+92+76=263\ \ c=\frac{263^2}{20}=3458.45\\
  \sum x^2=1160+1430+1000=3590\ \ TSS=3590-C=131.55\\
  BSS=\frac{95^2}{8}+{92^2}{6}+{76^2}{6}-C=3501.46-C=43.01\\
  WSS=131.55-43.01=88.54$

  \begin{description}

   \item[(i)]

  \hspace*{30mm}anova Table

  \begin{tabular}{cccc}
   Source&df&ss&ms\\
   \hline
   Between groups&2&43.01&21.505\\
   Within groups&17&88.54&5.208=$\hat{\sigma}^2(a)$\\
   \hline
   total&19&131.55
   \end{tabular}

   \item[(ii)]
    $H_0:\mu_M=\mu_T=\mu_G\ \ H_1:$ Not all equal $\alpha=5\%\\
    F_{2,17}=\frac{21.505}{5.208}=4.13$ significant at 5\%

   \item[(iii)]
    $\overline{x}_m=\frac{95}{8}=11.875\ \ \overline{x}_T=15.33\ \
     95\% CI$

    \begin{eqnarray*}
    -3.46&\pm& t_{17}\sqrt{5.208(\frac{1}{8}+\frac{1}{6})}\\
    -3.46&\pm&2.11\times 1.2325\\
    -3.46 &\pm& 2.60
    \end{eqnarray*}

  \end{description}


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