Question

In 1974, the Bellevue-Stratford Hotel in Philadelphia was the scene of an outbreak of what later became known as legionnaires’ disease. The cause of the disease was finally discovered to be bacteria that thrived in the air-conditioning units of the hotel. Owners of the Rip Van Winkle Motel, hearing about the Bellevue-Stratford, replaced their air-conditioning system. The following data are the bacteria counts in the air of eight rooms, before and after a new air-conditioning system was installed (measured in colonies per cubic foot of air). Has the new system succeeded in lowering the bacterial count? Base your analysis on a confidence interval.

Answer 1

Yes, we can say that there is statistical evidence that the new system succeeded in lowering the bacterial count.

Explanation:

The question is incomplete. The data for the rooms is:

Room Before After

121 11.8 10.1

163 8.2 7.2

125 7.1 3.8

264 14.0 12.0

233 10.8 8.3

218 10.1 10.5

324 14.6 12.1

325 14.0 13.7

We can calculate the difference in colonies after and before, and analyze this sample.

The difference is for each room:

Room Difference

121 -1.7

163 -1.0

125 -3.3

264 -2.0

233 -2.5

218 0.4

324 -2.5

325 -0.3

The sample size is n=8.

The sample mean is M=-1.6 and the standard deviation is s=1.24.

Then, the degrees of freedom are

`df=n-1=8-1=7`

We will calculate a 95% confidence interval for the difference in colonies. For a 95% CI and 7 degrees of freedom, the critical valule of t is t=2.365.

Then, the margin of error is:

`E=t\cdot s/√(n)=2.365*1.24/√(8)=2.9326/2.83=1.036`

Then, the lower and upper bounds of the CI are:

`LL=M-t\cdot s/√(n)=-1.6-1.036=-2.636\\\\UL=M+t\cdot s/√(n)=-1.6+1.036=-0.564`

With these CI we can conclude that the actual difference in the bacterial count in the rooms is, with 95% confidence, between -0.564 and -2.636. This also indicates that there is only 2.5% confidence that the difference in bacterial count is bigger than -0.564. So there is less than 2.5% that the bacterial count does not succeed (bacterial count equal or higher than 0).

This means that we can say that there is statistical evidence that the new system succeeded in lowering the bacterial count.