Six measurements were made of the magnesium ion concentration (in parts per million, o in a city's municipal water supply, with the following results. It is reasonable to assume th population is approximately normal.
$\begin{array}{llllll}
\hline 170 & 201 & 199 & 202 & 173 & 153 \
\hline
\end{array}$

Based on a $95 \%$ confidence interval for the mean magnesium ion concentration, is it reasonable to believe that the mean magnesium ion concentration may be greater than 208 ? (Hint: you should first calculate the $95 \%$ confidence interval for the mean magnesium ion concentration.) )

Question

Six measurements were made of the magnesium ion concentration (in parts per million, o in a city's municipal water supply, with the following results. It is reasonable to assume th population is approximately normal.
$\begin{array}{llllll}
\hline 170 & 201 & 199 & 202 & 173 & 153 \
\hline
\end{array}$

Based on a $95 \%$ confidence interval for the mean magnesium ion concentration, is it reasonable to believe that the mean magnesium ion concentration may be greater than 208 ? (Hint: you should first calculate the $95 \%$ confidence interval for the mean magnesium ion concentration.) )

Quizplus · Accepted Answer

To determine if it's reasonable to believe that the mean magnesium ion concentration may be greater than 208, we first calculate the sample mean ($\bar{x}$) and the sample standard deviation (s) of the given data. Then, we use these values to calculate the 95% confidence interval for the mean.The sample mean ($\bar{x}$) is calculated as:$$\bar{x} = \frac{170 + 201 + 199 + 202 + 173 + 153}{6} = \frac{1098}{6} = 183$$The sample standard deviation (s) is calculated as:$$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{(170-183)^2 + (201-183)^2 + (199-183)^2 + (202-183)^2 + (173-183)^2 + (153-183)^2}{5}}$$$$s = \sqrt{\frac{169 + 324 + 256 + 361 + 100 + 900}{5}} = \sqrt{\frac{2110}{5}} = \sqrt{422} \approx 20.54$$For a 95% confidence interval, the critical value (t*) for a t-distribution with 5 degrees of freedom (n-1 = 6-1 = 5) is approximately 2.571 (this value can vary slightly depending on the source, but it's around this value for a 95% confidence interval).The margin of error (E) is calculated as:$$E = t* \times \frac{s}{\sqrt{n}} = 2.571 \times \frac{20.54}{\sqrt{6}} \approx 2.571 \times 8.39 \approx 21.57$$The 95% confidence interval for the mean is:$$\bar{x} \pm E = 183 \pm 21.57 = (161.43, 204.57)$$Since the interval is (161.43, 204.57), it does not include 208. Therefore, it is not reasonable to believe that the mean magnesium ion concentration may be greater than 208 based on this confidence interval.

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Six Measurements Were Made of the Magnesium Ion Concentration (In $\begin{array}{llllll}\hline 170 & 201 & 199 & 202 & 173 & 153 \\\hline\end{array}$