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Given:

\(\begin{array}{l}m = 5\\n = 7\\\alpha = 0.01\end{array}\)

Given claim: Larger for the polluted region
The null hypothesis states that there is no difference. The alternative hypothesis states the given claim:

\(\begin{array}{l}{H_0}:{\mu _1} = {\mu _2}\\{H_a}:{\mu _1} < {\mu _2}\end{array}\)

Determine the rank of every data value. The smallest value receives the rank l, the second smallest value receives the rank 2, the third smallest value receives the rank 3, and so on.

If multiple data values have the same value, then their rank is the average of the corresponding ranks.

\({W_1}\)is the sum of the ranks in the smaller sample:

\(w = 25\)

The\(w\)-value corresponding to an upper-tailed test is then the largest possible rank sum decreased by the\(x\)-value corresponding to a lower-tailed test:

\(\begin{array}{l}w = m(n + m + 1) - 25\\w = 5(5 + 7 + 1) - 25\\w = 65 - 25\\w = 40\end{array}\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true.
The P-value is the number (or interval) in the column\({P_0}(W \ge c)\)of the Wilcoxon rank-sum test table in the appendix that contains the w-value in the row :

\(\begin{array}{l}m = 5,\\n = 7,\\c = 40\end{array}\)

\(P > 0.053\)

If the probability (P-value) is less than the significant level\(\alpha \), then reject the null hypothesis:

\(P > 0.01 \Rightarrow \)Fail to reject \({H_0}\).

Therefore, there is not sufficient evidence to support the claim that the true average fluoride concentration for livestock grazing in the polluted region is larger than for the unpolluted region.