Step 1 :The null and alternative hypotheses in symbolic form for this claim are: \(H_{0}: \mu_{d} \geq 0\) and \(H_{a}: \mu_{d} < 0\)
Step 2 :The significance level is \(\alpha=0.1\)
Step 3 :Calculate the differences between the zinc concentrations in the bottom and surface water for each location, then find the mean and standard deviation of these differences. The differences are: .015, .028, .177, .121, .102, .107, .019, .066, .058, .111. The mean of these differences is \(\bar{d} = .0804\). The standard deviation of these differences is \(s_d = 0.055\)
Step 4 :Calculate the test statistic as follows: \(t = \frac{\bar{d} - 0}{s_d / \sqrt{n}} = \frac{.0804 - 0}{.055 / \sqrt{10}} = 5.503\) (rounded to 3 decimal places)
Step 5 :The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. Since we are conducting a one-tailed test, we look up the p-value corresponding to our test statistic in a t-distribution table with n-1 = 10-1 = 9 degrees of freedom. The p-value is less than 0.0001 (rounded to 4 decimal places)
Step 6 :Since the p-value is less than the significance level (0.0001 < 0.1), we reject the null hypothesis
Step 7 :\(\boxed{\text{Therefore, there is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source.}}\)