Question

The article "Occurrence and Distribution of Ammonium in Iowa Groundwater" (K. Schilling, Water Environment Research, 2002:177–186) describes measurements of ammonium concentrations (in mg/L) at a large number of wells in the state of Iowa. These included 349 alluvial wells and 143 quaternary wells. Of the alluvial wells, 182 had concentrations above 0.1, and 112 of the quaternary wells had concentrations above 0.1. Find a 95% confidence interval for the difference between the proportions of the two types of wells with concentrations above 0.1.

Answer 1

Answer: 95% confidence interval for the difference between the proportions would be (1.31, 1.39).

Explanation:

Since we have given that

Number of alluvial wells = 349

Number of quaternary wells = 143

Number of alluvial wells that had concentrations above 0.1 = 182

Number of quaternary wells that had concentrations above 0.1 = 112

Average of alluvial wells = 0.27

Standard deviation = 0.4

Average of quaternary wells = 1.62

Standard deviation =1.70

So, 95% confidence interval gives

alpha = 5% level of significance.

`(\alpha)/(2)=2.5\%\\\\z_{(\alpha)/(2)}=1.96`

So, 95% confidence interval becomes,

`(1.62-0.27)\pm 1.96\sqrt{(0.4^2)/(349)+(1.7^2)/(143)}\\\\=1.35\pm 1.96* 0.020\\\\=(1.35-0.040,1.35+0.040)\\\\=(1.31,1.39)`

Hence, 95% confidence interval for the difference between the proportions would be (1.31, 1.39).