A Farmer Has Five Different Fields $\mu 1$ Be the Mean for Fertilizer

Essay

A farmer has five different fields. He wishes to test two different fertilizers. The yields of corn may vary from field to field depending on a number of factors such as soil quality, sunlight etc. He divides each field into three sections. For each field, he randomly assigns one of the sections to receive fertilizer A, one of the sections to receive fertilizer B, and one of the sections to receive no fertilizer. At harvest time, he records the corn yields in bushels per acre for each section of each field. The results are shown in the table. Let $\mu 1$ be the mean for fertilizer $\mathrm { A } , \mu _ { 2 }$ the mean for fertilizer $\mathrm { B }$ , and $\mu _ { 3 }$ the mean for no fertilizer. In a test of $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 }$ versus $\mathrm { H } _ { 1 }$ : at least one of the means is different, the null hypothesis is rejected at the $\alpha = 0.05$ level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha = 0.05$ . Give the P-value for each of the pairwise tests and state your conclusion.
$\begin{array}{l|c|c|c} & \text { Fertilizer A } & \text { Fertilizer B } & \text { No Fertilizer } \\\hline \text { Field 1 } & 132 & 141 & 128 \\\text { Field 2 } & 136 & 135 & 130 \\\text { Field 3 } & 144 & 147 & 139 \\\text { Field 4 } & 126 & 134 & 122 \\\text { Field 5 } & 130 & 128 & 125\end{array}$