Chapter PVI, Problem 23RE

To determine

To explain has the new system succeeded in lowering the bacterial count and base your analysis on a confidence interval.

Answer to Problem 23RE

We are $95\%$ confident for the difference, before minus after, is $(0.58,2.65)$ counts and since the entire interval is above zero these data suggest that the new air-conditioning system was effective in reducing average bacteria counts.

Explanation of Solution

It is given in the question that the Bellevue-Stratford Hotel in Philadelphia was the scene of an outbreak of what later became known as legionnaires’ disease. Let us first check the assumptions and conditions for inference as:

Matched pairs: It is satisfied as data are matched pairs before and after for the same rooms.

Random condition: It is satisfied as the rooms are randomly selected.

  $10\%$ less condition: It is satisfied as the sample is less than $10\%$ of all sample.

Normal condition: It is satisfied because histograms shows that differences are roughly uni-modal and symmetric with no outliers.

So, all the conditions and assumptions are satisfied. Thus, to find the confidence interval for mean we will use the calculator $\text{TI}-89$ . In the STAT Intsmenu, choose $4:2-$ SampTInt. You must specify if you are using data stored in two lists or if you will enter the means, standard deviations, and sizes of both samples. You must also indicate whether to pool the variances (when in doubt, say no) and specify the desired level of confidence. So, we have the list as:

Room number	Before	After	Difference
121	11.8	10.1	1.7
163	8.2	7.2	1
125	7.1	3.8	3.3
264	14	12	2
233	10.8	8.3	2.5
218	10.1	10.5	-0.4
324	14.6	12.1	2.5
325	14	13.7	0.3

And $\alpha =0.05$ so by using the calculator $\text{TI}-89$ , the confidence interval will be as:

  $(0.58,2.65)$ .

So, we are $95\%$ confident for the difference, before minus after, is $(0.58,2.65)$ counts. Since the entire interval is above zero these data suggest that the new air-conditioning system was effective in reducing average bacteria counts.