Problem 11SE
In the article "Groundwater Electromagnetic Imaging in Complex Geological and Topographical Regions: A Case Study of a Tectonic Boundary in the French Alps" (S. Houtot, P. Tarits, et al. Geophysics, 2002:10481060), the pH was measured for several water samples in various locations near Gittaz Lake in the French Alps. The results for 11 locations on the northern side of the lake and for 6 locations on the southern side are as follows:
Northern side:

8.1 
8.2 
8.1 
8.2 
8.2 
7.4 
7.3 
7.4 
8.1 
8.1 
7.9 


Southern side: 
7.8 
8.2 
7.9 
7.9 
8.1 
8.1 
Find a 98% confidence interval for the difference in pH between the northern and southern side.
Answer:
Step 1 of 1:
In this question, we are asked to find a 98% confidence interval for the difference in between the northern and southern side.
The results for 11 locations on the northern side of the lake and for 6 locations on the southern side are given below on the table:
Northern side:

8.1 
8.2 
8.1 
8.2 
8.2 
7.4 
7.3 
7.4 
8.1 
8.1 
7.9 


Southern side: 
7.8 
8.2 
7.9 
7.9 
8.1 
8.1 
Let be a small sample of size from a population with mean , let be a small sample of size from a population with mean, then a level confidence interval for difference between population mean is
……….(1)
Has an approximate Student’s distribution.
The number of degrees of freedom to use for this distribution is given by
……….(2)
Let be a sample. The sample mean is
……….(3)
Let be a sample. The sample variance is
……….(4)
Use equation (3) and (4) to calculate the sample mean and sample standard deviation.
Let be a sample taken on 11 locations on the northern side of the lake.
Let be a sample taken on 6 locations on the southern side of the lake.
Hence,
sample mean
Sample variance
sample mean
Sample variance
Using above value, calculate the degrees of freedom,
=
(rounded down to nearest integer)
So the confidence interval is using equation (1),
= 0.091 0.3279
Hence a 98% confidence interval for the difference in between the northern and southern side is