Q. 70 - The Practice of Statistics for AP

Question

Trace metals found in wells affect the taste of drinking water, and high concentrations can pose a health risk. Researchers measured the concentration of zinc (in milligrams/liter) near the top and the bottom of $10$ randomly selected wells in a large region. The data are provided in the table below. 

(a) Construct and interpret a $95\%$ confidence interval for the mean difference $\mu$ in the zinc concentrations from these two locations in the wells.

(b) Does your interval in part (a) give convincing evidence of a difference in zinc concentrations at the top and bottom of wells in the region? Justify your answer. 

Step-by-Step Solution

Answer

(a) We are $95\%$ confident that the true population mean difference is between $0.0430$ and $0.1178$.

(b) There is sufficient evidence to support the claim of a difference in zinc concentrations at the top and bottom of wells in the region.

Part(a) Step 1: Given Information

$\begin{array}{|rrr|}\hline \text{ Sample 1 }& \text{ Sample 2 }& \text{ Difference D }\\ 0.43& 0.415& 0.015\\ 0.266& 0.238& 0.028\\ 0.567& 0.39& 0.177\\ 0.531& 0.41& 0.121\\ 0.707& 0.605& 0.102\\ 0.716& 0.609& 0.107\\ 0.651& 0.632& 0.019\\ 0.589& 0.523& 0.066\\ 0.469& 0.411& 0.058\\ 0.723& 0.612& 0.111\\ \hline\end{array}$

Mean $=0.0804$

Sd  $=0.0523$

Part(a) Step 2: Explanation

The mean is the sum of all values divided by the number of values:

$$\begin{gathered} \overline{x}=\frac{0.015+0.028+\dots +0.058+0.111}{10} \\ \approx 0.0804 \end{gathered}$$

$n$ is the number of values in the data set.

The variance is the sum of squared deviations from the mean divided by $n-1$ :

$$\begin{gathered} s^2=\frac{(0.015-0.0804{)}^2+\dots .+(0.111-0.0804{)}^2}{8-1} \\ \approx 0.0025 \end{gathered}$$

The standard deviation is the square root of the variance:

$$\begin{gathered} s=\sqrt{0.0025} \\ \approx 0.0523 \end{gathered}$$

Part(a) Step 3: Calculation

Determine the t-value by looking in the row starting with degrees of freedom $n-1=10-1=9$ and in the row with $c=95\%$ in table B:

$t^{*}=2.262$

The margin of error is then:

$$\begin{gathered} E=t^{*}·\frac{s}{\sqrt{n}} \\ =2.262\times \frac{0.0523}{\sqrt{10}} \\ \approx 0.0374 \end{gathered}$$

Then the confidence interval becomes:

$$\begin{gathered} 0.0430=0.0804-0.0374 \\ =\overline{x}-E<\mu <\overline{x}+E \\ =0.0804+0.0374 \\ =0.1178 \end{gathered}$$

Part(b) Step 1: Given Information

$\begin{array}{|rrr|}\hline \text{ Sample 1 }& \text{ Sample 2 }& \text{ Difference D }\\ 0.43& 0.415& 0.015\\ 0.266& 0.238& 0.028\\ 0.567& 0.39& 0.177\\ 0.531& 0.41& 0.121\\ 0.707& 0.605& 0.102\\ 0.716& 0.609& 0.107\\ 0.651& 0.632& 0.019\\ 0.589& 0.523& 0.066\\ 0.469& 0.411& 0.058\\ 0.723& 0.612& 0.111\\ \hline\end{array}$

Mean $=0.0803$

Sd $=0.0523$

Part(b) Step 2: Explanation

The mean is the sum of all values divided by the number of values:

$$\begin{gathered} \overline{x}=\frac{0.015+0.028+\dots +0.058+0.111}{10} \\ \approx 0.0804 \end{gathered}$$

$n$ is the number of values in the data set.

$n=10$

The variance is the sum of squared deviations from the mean divided by $n-1$ :

$$\begin{gathered} s^2=\frac{(0.015-0.0804{)}^2+\dots ..+(0.111-0.0804{)}^2}{8-1} \\ \approx 0.0025 \end{gathered}$$

The standard deviation is the square root of the variance:

$$\begin{gathered} s=\sqrt{0.0025} \\ \approx 0.0523 \end{gathered}$$

Part(b) Step 3: Calculation

Determine the t-value by looking in the row starting with degrees of freedom $n-1=10-1=9$ and in the row with $c=95\%$ in table B:

$t^{*}=2.262$

The margin of error is then:

$$\begin{gathered} E=t^{*}·\frac{s}{\sqrt{n}} \\ =2.262\times \frac{0.0523}{\sqrt{10}} \\ \approx 0.0374 \end{gathered}$$

Then the confidence interval becomes:

$$\begin{gathered} 0.0430=0.0804-0.0374 \\ =\overline{x}-E<\mu <\overline{x}+E \\ =0.0804+0.0374 \\ =0.1178 \end{gathered}$$

We are $95\%$ confident that the true population mean difference is between $0.0430$ and $0.1178$.

The confidence interval does not contain $0$, which indicates that there is a difference between the population means and thus there is sufficient evidence to support the claim of a difference in zinc concentrations at the top and bottom of wells in the region.