Effects of Acid Rain

This question was administered in 92 December.

Acid Rain is a serious problem in Canada. As acidification progresses, the eco-system simplifies, i.e., the number of different species of fish, plant life, insects declines, until eventually, the lake is virtually sterile - no fish and few water plants. To investigate the effects of acid rain, an experiment has been running for the last decade in the Experimental Lakes Area (ELA), a series of small lakes just north of Kenora, Ontario about 250 km east of Winnipeg. These experiments have been instrumental in helping scientists understand the processes involved in acidification and have won world wide acclaim. [You might be interested to know that, as usual, the response of Canadian governments has been to cut funding for this very successful experimental program!] The following questions are based upon actual experiments conducted at the lakes.

A series of 20 small lakes (each less than 10 ha in area) were selected. These lakes were randomly assigned to either the control group or the treatment group. The treatment group was acidified by dumping sulfuric acid into the lakes at rates comparable to what would occur during acid rain. After one year, the lakes were sampled and measures of diversity were computed for each lake. Higher values of the diversity imply more types of living organisms occur which indicates, in general, a more healthy lake.

Here is the data:

Treatment Lakes	Control Lakes
52	49
54	61
39	52
48	50
37	49
47	45
37	68
54	53
45	69
46	72
Summary statistics Mean = 45.900 Std Deviation = 6.5056	Summary statistics Mean = 56.800 Std Deviation = 9.8184

1. Why did we randomize the lakes to either the treatment or the control group? Why didn't we apply the treatment to all of the lakes?
   

   We randomize so that the effects of other, uncontrolled factors will be roughly equal in both groups. Consequently, and difference we see, may be attributable to the treatment.

   We need controls to obtain a baseline against which we can compare the treatment group. For example, the diversity of the lakes may change naturally over time - without a control group, we cannot conclude that any change must be due to the treatment.

2. Using the summary statistics above, construct a 95% confidence interval for the mean diversity of the control lakes. Interpret this interval. Plot the confidence interval on the box-plot on the next page.

   ± t_n-1 s/ = 56.8 ± (2.2622)(9.8184)/sqrt(10) = 56.8 ± 7.01 = (49.7->63.9)

   We are 95% confident that the true mean diversity index for ALL untreated lakes is in this range.
   
   

3. What is the difference between a box-plot and a confidence interval plot?
   

   A box-plot displays statistics about INDIVIDUAL data values in the sample.

   The confidence interval plot displays plausible values for the POPULATION MEAN.

4. Based upon the above plots, does there appear to be a difference between the mean diversity of control and experimental lakes? Explain how you came to this conclusion.

   It is hard to tell since the two confidence intervals overlap a bit. The box-plots are irrelevant to the question.

5. Compute a 95% confidence interval for the difference in the mean diversity of the control and experimental lakes. Interpret this interval. Based upon this interval, does there appear to be a difference in the mean diversity? Why?
   
   
   

   We are 95% confident that the true difference in the mean diversity index lies within this range.

   Since the interval does not include 0, we conclude that there is evidence of a difference in the mean diversities.

6. Consider the control lakes. Explain the difference between _control and µ_control and how the c.i. links the two.

   _control is the sample mean computed from the sample data
   

   µ_control is the population mean which is unknown.

   The confidence interval gives a range of plausible values for the population mean (µ_control) computed from the sample statistics.