Questions: 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. Location Zinc concentration in bottom water Zinc concentration in surface water --------- 1 .430 .415 2 .266 .238 3 .567 .390 4 .531 .410 5 .707 .605 6 .716 .609 7 .651 .632 8 .589 .523 9 .469 .411 10 .723 .612 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 mud=mu1-mu2, where mu1 is the population mean zinc concentration in the bottom of water and mu2 is the mean zinc concentration in the surface of water. H0: mud Select an answer 0 Ha: mud Select an answer 0 b. What is the significance level? alpha= c. What is the test statistic? Round to 3 decimal places. 20= d. What is the p-value? Round to 4 decimal places. p= e. Make a decision. O Do not reject the null Reject the null f. What is the conclusion? There is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source. 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.

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. Location Zinc concentration in bottom water Zinc concentration in surface water --------- 1 .430 .415 2 .266 .238 3 .567 .390 4 .531 .410 5 .707 .605 6 .716 .609 7 .651 .632 8 .589 .523 9 .469 .411 10 .723 .612 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 mud=mu1-mu2, where mu1 is the population mean zinc concentration in the bottom of water and mu2 is the mean zinc concentration in the surface of water. H0: mud Select an answer 0 Ha: mud Select an answer 0 b. What is the significance level? alpha= c. What is the test statistic? Round to 3 decimal places. 20= d. What is the p-value? Round to 4 decimal places. p= e. Make a decision. O Do not reject the null Reject the null f. What is the conclusion? There is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source. 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.
Transcript text: 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. Location | Zinc concentration in bottom water | Zinc concentration in surface water ---|---|--- 1 | .430 | .415 2 | .266 | .238 3 | .567 | .390 4 | .531 | .410 5 | .707 | .605 6 | .716 | .609 7 | .651 | .632 8 | .589 | .523 9 | .469 | .411 10 | .723 | .612 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 bottom of water and $\mu_{2}$ is the mean zinc concentration in the surface of water. \[ \begin{array}{l} H_{0}: \mu_{d} \text { Select an answer } 0 \\ H_{a}: \mu_{d} \text { Select an answer } 0 \end{array} \] b. What is the significance level? $\alpha=$ $\qquad$ c. What is the test statistic? Round to 3 decimal places. $20=$ $\square$ d. What is the $p$-value? Round to 4 decimal places. $p=$ $\square$ e. Make a decision. O Do not reject the null Reject the null f. What is the conclusion? There is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source. 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.
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Solution

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Solution Steps

Step 1: Hypotheses

We set up the null and alternative hypotheses as follows: \[ H_0: \mu_d = 0 \quad \text{(The mean difference in zinc concentration is zero)} \] \[ H_a: \mu_d > 0 \quad \text{(The mean difference in zinc concentration is greater than zero)} \]

Step 2: Significance Level

The significance level is given as: \[ \alpha = 0.1 \]

Step 3: Test Statistic Calculation

The standard error (SE) is calculated as: \[ SE = \frac{0.0523}{\sqrt{10}} = 0.0165 \] The test statistic \( t \) is calculated using the formula: \[ t = \frac{0.0804}{0.0165} = 4.8638 \]

Step 4: Critical Value

For a two-tailed test at \( \alpha = 0.1 \), the critical value is: \[ t_{\alpha/2, df} = t_{(0.05, 9)} = 1.8331 \]

Step 5: p-value Calculation

The p-value is calculated as: \[ P = 2 \times (1 - T(|t|)) = 2 \times (1 - T(4.8638)) = 0.0009 \]

Step 6: Decision

Since the p-value \( 0.0009 \) is less than the significance level \( \alpha = 0.1 \), we reject the null hypothesis.

Step 7: Conclusion

There is sufficient evidence to support the claim that the zinc concentration is less on the surface than the bottom of the water source.

Final Answer

\[ \boxed{\text{Reject the null hypothesis; there is sufficient evidence to support the claim.}} \]

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