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QUESTION Eight golfers play a round of golf on two consecutive
Saturdays. On the first Saturday they returned scores of
72,89,69,70,85,71,96,86 and on the second Saturday in the same
order 72,80,71,70,82,72,90,84.

\begin{description}
\item[(a)]
Assuming that the differences in their scores are drawn from a
normal population, is there significant evidence that their golf
has improved?

\item[(b)]
Carry out the appropriate test of the scores for the second
Saturday had been given to you in a different and unknown order.
\end{description}



ANSWER

\begin{tabular}{cccccccc}
  72&82&69&70&85&71&96&86\\
  72&80&71&70&82&72&90&84
  \end{tabular}
  $H_0:\mu_1=\mu_2\ \ H_1:\mu_1>\mu_2\ \ \alpha =5\%$

  \begin{description}
   \item[(a)]
    assuming paired sample data\\
    \begin{tabular}{ccccccccc}
    d&0&2&-2&0&3&-1&6&2
    \end{tabular}
    $H_0:\mu_d=0\ \ H_1:\mu_d \neq 0$\\
    Test 4a, Paired sample, two means.
    $z=\frac{\overline{d}-0}{\frac{s_d}{\sqrt{n}}}\sim t_n\\
    \overline{d}=1.25\ \ s_d=2.5495\ \ n=8$

    $\begin{array}{l}
     z=\frac{1.25}{\frac{2.5495}{\sqrt{8}}}=1.39\\
    \textrm{is not significant.}\\
    \textrm{Hence accept }H_0.
    \end{array}
    \ \ \
    \begin{array}{c}
    \epsfig{file=641-5-4.eps, width=60mm}
    \end{array}$


   \item[(b)]
   Assuming independent data
   \begin{eqnarray*}
   \overline{x}_1 & = & 78.875\\
   s_1 & = & 9.8334\\
   overline{x}_2 & = & 77.625\\
   s_2 & = & 7.4054\\
   n_1 & = & n_2=8
   \end{eqnarray*}
   Test 4, assume normal distribution, variances equal.
   $$z=\frac{\overline{x}_1-\overline{x}_2}{\sqrt{\{s^2(\frac{1}{n_1}
   +\frac{1}{n_2})\}}}$$
   $$s^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}\sim tn_1+n_2-2$$
   \begin{eqnarray*}
   s & = & 8.7045\\
   z & = & \frac{1.25}{8.7045\sqrt{\frac{1}{8}+\frac{1}{8}}}=0.29
   \end{eqnarray*}
   Clearly
   not significant as $t_{14}$ hence accept $H_0$. Test in (b)
   much less sensitive.

  \end{description}

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