$0 / 1$ pt $O_{2} \rightleftarrows$
Trace metals in drinking water affect the flavor and an unusually high concentration can pose a health hazard. Ten pairs of data were taken measuring zinc concentration in bottom water and surface water of a water source.
\begin{tabular}{|c|c|c|}
\hline Location & \begin{tabular}{c}
Zinc \\
concentration in \\
bottom water
\end{tabular} & \begin{tabular}{c}
Zinc \\
concentration in \\
surface water
\end{tabular} \\
\hline 1 & .430 & .415 \\
\hline 2 & .266 & .238 \\
\hline 3 & .567 & .390 \\
\hline 4 & .531 & .410 \\
\hline 5 & .707 & .605 \\
\hline 6 & .716 & .609 \\
\hline 7 & .651 & .632 \\
\hline 8 & .589 & .523 \\
\hline 9 & .469 & .411 \\
\hline 10 & .723 & .612 \\
\hline
\end{tabular}
Do the data support that the zinc concentration is less on the surface than the bottom of the water source, at the $\alpha=0.1$ level of significance? Note: A normal probability plot of difference in zinc concentration between the bottom and surface of water indicates the population could be normal and a boxplot indicated no outliers.
a. Express the null and alternative hypotheses in symbolic form for this claim. Assume $\mu_{d}=\mu_{1}-\mu_{2}$, where $\mu_{1}$ is the population mean zinc concentration in the botiom of water and $\mu_{2}$ is the mean zinc concentration in the surface of water.
\[
\begin{array}{l}
H_{0}: \mu_{\bar{d}} \text { Select an answer } \vee \\
H_{a}: \mu_{d} \text { Select an answer } \vee
\end{array}
\]
b. What is the significance level?
\[
\alpha=
\]
c. What is the test statistic? Round to 3 decimal places.
\[
? \vee=
\]
d. What is the $p$-value? Round to 4 decimal places.
\[
p=
\]
e. Make a decision.
Do not reject the null
Reject the null
f. What is the conclusion?
There is not sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source.
There is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source.
Gihmit Ruactinn
Answer
\(\boxed{\text{Therefore, there is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source.}}\)
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 ：\(\boxed{\text{Therefore, there is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source.}}\)
