The concentration of hexane (a common solvent) was measured in units of micrograms per liter for a simple random sample of twelve specimens of untreated ground water taken near a municipal Landfill. The sample mean was 720.2 with a sample standard deviation of 8.8. Eleven specimens of Treated ground water had an average hexane concentration of 695.1 with a standard deviation of 9.1.
It is reasonable to assume that both samples come from populations that are approximately normal. Construct A 90% confidence interval for the reduction of hexane concentration after treatment.
A) (23.6, 26.6)
B) (18.3, 31.9)
C) (21.8, 28.4)
D) (22.2, 28.0)

Question

Quizplus · Accepted Answer

The confidence interval for the difference between two means can be calculated using the formula for the confidence interval of the difference between two independent means, assuming normal distributions and unknown but assumed equal variances. The formula for a 90% confidence interval is $(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$, where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $s_1^2$ and $s_2^2$ are the sample variances, $n_1$ and $n_2$ are the sample sizes, and $t_{\alpha/2}$ is the t-value for the desired confidence level with degrees of freedom approximated by the smaller of $n_1-1$ and $n_2-1$. Given the sample sizes and standard deviations, and using the appropriate t-value for a 90% confidence interval with degrees of freedom close to the smaller sample size minus one (which would be 10 for eleven specimens), the calculation results in a confidence interval that suggests choice B is correct, indicating the estimated range of reduction in hexane concentration after treatment.

The Concentration of Hexane (A Common Solvent) Was Measured in Units