Example
At an agriculture station it was decided to test the effect the given fertilizer on wheat production. To accomplish this, 24 plots of having equal area were chosen , half of these were trated with the fertilizer and other half were untreated.Otherwise the conditions were same. The mean yeild of wheat on the untreated plots was 4.8 bushels with a standard deviation of 0..40 bushels , while the mean yeild on the treated plots bushels with a standard deviation of 0.36 bushels.Can we conclude that there is significant improvement in wheat production because of fertilizer if a significance level of (a) 0.1 (b) 5% is used .(c) what is the P value of the test?
Solution
if \[\mu _1\] and\[\mu _2\] denote population mean yeilds of wheat on treated and untreated land, respectively , we have to decide between the hypothesis
\[H_0\]: \[\mu _1=\mu _2, \] and the difference is due to chance
\[H_1\]: \[\mu _1 > \mu _2, \] and the fertilizer improves the yeild.
Under the hypothesis \[H_0\],
\[T=\frac{X-Y}{\sigma \sqrt{( \frac{1}{n_1}+\;\frac{1}{n_2})}}\]
where \[\sigma = \sqrt{\frac{ms_1^2+ns_2^2}{n+m-2}}\]
In this case \[ \sigma = \sqrt{\frac{12(0.40)^2+12(0.36)^2}{12+12-2}}=0.397\]
\[ T=\frac{5.1-4.8}{0.397 \sqrt{( \frac{1}{12}+\;\frac{1}{12})}}=1.85\]
(a) On the basis of one- tailed test at 0.01 level of significance , we could reject \[H_0\] if T were greater than \[ t_{0.99}\]
which , for \[ n_1+n_2-2=22\] degrees of freedom , is 2.51
therefore we can not reject \[H_0\] at 0.01 level of significance .
(b) On the basis of one- tailed test at 0.05 level of significance , we could reject \[H_0\] if T were greater than \[ t_{0.95}\]
which , for \[ n_1+n_2-2=22\] degrees of freedom , is 1.72
therefore we can reject \[H_0\] at 0.05 level of significance
we conclude that the improvement in yeild of wheat by use of fertilizer is probably significant.However before definite conclusions are drawn concerning the usefullness of the fertilizer , it may be desirable to have some further evidence
(c) The P value is P(T\[\geq \]1.85)
Assumption
\[n_1=m, n_2=n,s_1=s,s_2=r \]
and \[ \sigma =K\]
