Reference no: EM132483916

Air pollution is a serious issue in many places, including here in Cincinnati, as you will learn by watching the video on the left. Particulate matter (PM), consisting of tiny particles in the air, is a kind of air pollution suspected of causing respiratory illness. PM has many possible sources, such as ashes and tiny pieces of rubber from automobile and truck tires.

One town that has experienced high levels of PM, especially in the winter, is Libby, Montana. This is because many houses there are heated by wood stoves, which produce a lot of particulate pollution. In 2006 and early 2007, a program was undertaken to reduce the air pollution by replacing every wood stove with a newer, cleaner-burning model. As part of a study to compare the amount of pollution before and after the stove replacement, the PM level was measured after the stoves were replaced on a sample of 20 days in the winter of 2007-2008. These are the results in micrograms per cubic meter:

21.7, 27.8, 24.7, 15.3, 18.4, 14.4, 19.0, 23.7, 22.4, 25.6, 15.0, 17.0, 23.2, 17.7, 11.1, 29.8, 20.0, 21.6, 14.8, 21.0

As you can see from this data, the amount of PM varies from day to day. Say we would like to estimate the population mean, which is the mean over all days of the winter of 2007-2008. The best choice of a single number would be the sample mean, which is 20.21. We call this a point estimate.

The problem with point estimates is that they are rarely exactly equal to the true value that they are estimating. In this example, it is more likely that the mean level of PM for all of 2007-2008 is somewhat more or less than the sample mean, 20.21. For the point estimate to be useful, we need to describe just how close to the true value it is likely to be. Statisticians construct confidence intervals. A confidence interval is a range of values that is likely to contain the true value being estimated. One of the benefits of confidence intervals is that they come with a measure of the level of confidence we can have that the true value is contained within the interval. For this module problem you will:

1. Show that we can be 95% confident that the true mean, the population mean, is in the interval 20.21 ± 2.27, i.e. there is a 95% likelihood that it is between 17.94 and 22.48. (Essential Statistics, Monk).

2. Find a 99% confidence interval. Choose an appropriate way to represent the intervals and interpret the results.

3. Discuss: Which interval gives us more confidence? Which interval is larger? More reliable? What confidence level would you use in this situation and why? What effect would a larger sample size have on the reliability of the results?