Question

An experiment was conducted to determine the concentration of a particular bacterium (Pseudomonas syringae) found adhering to rocks in river beds. Of particular interest was whether the number of bacteria was the same for rocks near the source of the river versus rocks near the outlet of the river. The experiment was conducted as follows. Six rivers in Iowa were randomly sampled. Then, for each river, the number of bacteria was measured for rocks at the source of the river and for rocks at the outlet of the river (and measured as number of bacteria per unit sample).

The data are provided below with some summary statistics:

River 1 2 3 4 5 6 Sample Mean Sample Variance

Source 5600 2600 3260 4910 3750 1720 3640 2075800

Outlet 5480 2380 3300 4800 3680 1600 3540 2107520

>round(quantile(diff.boot, probs=c(0.005, 0.01, 0.05, 0.10, 0.90, 0.95, 0.99, 0.995)), 3)

0.5% 1% 5% 10% 90% 95% 99% 99.5%

-2.776, -2.569, -1.677, -1.299, 1.910, 2.385, 7.906, 7.906

(a) Construct a 99% confidence interval for the difference in the mean number of bacteria at the source compared to the mean number of bacteria at the outlet. Assume whatever normality assumptions are required based on the sampling.

(b) Without doing any further work, comment on what conclusions you could draw if you conducted a test of the null hypothesis that the mean number of bacteria is the same at the source and at the outlet, versus the two-sided alternative.

(c) The scientists then decided instead to construct a 99% bootstrap T confidence interval. They’ve generated the bootstrap T distribution and a number of its quantiles. Construct a 99% bootstrap T confidence interval from the given information, explaining each step.

Answer 1