[Im am told to carry out a suitable hypothesis test to determine whether the removal of the red cedar trees has reduced the mean number of leaves with rust spots.
I have stated the null hypothesis which is the mean number of leaves with rusts sports had decreased and alternate hypothesis which is the mean number of leaves with rust spots did not decrease.
I have worked out the variance for both years which for year 1 is 538.5 and year 2 is 361.6429. I have also worked out the mean for both the years which for year 1 the mean is 46.75 and year 2 the mean is 36.25.
I think I have to use the formula T=((xbar)- µ0)/(σ/sqrtn). however I am not given the mean in the description given to me
You must use paired samples $t$-test.
The values of differences are found in the table: $$\begin{array}{c|c|c} \text{Tree}&Y1&Y2&d\\ \hline 1&38&32&6\\ 2&10&16&-6\\ 3&84&57&27\\ 4&36&28&8\\ 5&50&55&-5\\ 6&35&12&23\\ 7&73&61&12\\ 8&48&29&19\\ \hline &\bar{Y_1}=46.75&\bar{Y_2}=36.25&\bar{x}_d=10.5\\ &&&s_d=12.2 \end{array}$$
The hypothesis test: $$H_0:\mu_d=0\\ H_a:\mu_d>0\\ t=\frac{\bar{x}_d-0}{s_d/\sqrt{n}}=\frac{10.5}{12.2/\sqrt{8}}=2.43\\ t_{\alpha,n-1}=t_{0.05,7}=1.89\\ t>t_{\alpha} \Rightarrow \text{Reject $H_0$}.$$ Conclusion: There is a significant evidence for the removal of the red cedar trees reduces the mean number of leaves with rust spots.