Question

Methyl t-butyl ether (MTBE) is an organic water contaminant that often results from gasoline spills. The level of MTBE (in parts per billion) was measured for a sample of 12 well sites located near a gasoline service station in New Jersey (Environmental Science & Technology,January 2005). The data are listed in the accompanying table.

a. Give a point estimate for m, the true mean MTBE level for all well sites located near the New Jersey gasoline service station.

b. Calculate and interpret a 99% confidence interval for m.

c. What assumptions are required for the interval, part b, to be valid? Are these assumptions reasonably satisfied?

Short answer

1. The point estimate of m is 97.1667
2. The Confidence interval is$(-4.8245,199.1445)$
3. The sample data value must be independent of one another.

Step-by-step solution

(a) Given the information

$\begin{array}{c}\text{Pointestimatefor}\mathrm{\mu },\\ \mathrm{\mu }=\frac{\mathrm{Ex}}{\mathrm{n}}\\ =\frac{150+12+367+251+38+63+12+8+11+13+134+107+150+12}{12}\\ =97.1667\end{array}$

(b) 99% confidence interval

$\begin{array}{l}\mathrm{CI}=\overline{\mathrm{x}}\pm {\mathrm{t}}_{2/2},\mathrm{df}\frac{\mathrm{S}}{\sqrt{\mathrm{n}}}\\ \text{Fromsample,}\overline{\mathrm{x}}=\frac{\mathrm{Ex}}{\mathrm{n}}=97.16\end{array}$

$\mathrm{S}=\sqrt{\frac{\mathrm{E}{({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}})}^2}{\mathrm{n}-1}}=113.7595$

Confidence Level = 99% = 0.99

$\begin{array}{c}\text{L=1–C}\\ \text{=1}-\text{0.99}\\ \text{=0.01}\end{array}$

${\mathrm{t}}_{2/2},\mathrm{df}=\frac{{\mathrm{t}}_{0.01}}{2},12-1=3.1058$

Now, the Confidence interval for 99% is:

$\begin{array}{c}\mathrm{CI}=\overline{\mathrm{x}}\pm {\mathrm{t}}_{2/2},\mathrm{df}\frac{\mathrm{S}}{\sqrt{\mathrm{n}}}\\ =97.16\pm 3.1058\times \frac{113.75}{\sqrt{12}}\\ =(-4.8245,199.1445)\\ \mathrm{CI}=(-4.8245,199.1445)\end{array}$

(c) Valid assumptions. Assumptions reasonably satisfied

Every sample's data value must be independent of one another. The measured values in the two samples ought to be independent of one another. The size of the sample should be at least 30 or almost typical in form. The two distinct samples must be randomly selected as well as a representative of the general population.

Yes, the conditions for creating a confidence interval are fulfilled. The 90% confidence interval is$(-4.8245,199.1445)$ .