Question 7.1: The chemical composition of soil varies with depth. The arti...
The chemical composition of soil varies with depth. The article “Sampling Soil Water in Sandy Soils: Comparative Analysis of Some Common Methods” (M. Ahmed, M. Sharma, et al., Communications in Soil Science and Plant Analysis, 2001:1677-1686) describes chemical analyses of soil taken from a farm in western Australia. Fifty specimens were taken at each of the depths 50 and 250 cm. At a depth of 50 cm, the average· NO_3 concentration (in mg/L) was 88.5 with a standard deviation of 49.4. At a depth of 250 cm, the average concentration was 110.6 with a standard deviation of 51.5. Find a 95% confidence interval for the difference between the NO_3 concentrations at the two depths.
Let X_1 , … , X_{50} represent the concentrations of the 50 specimens taken at 50 cm, and let Y_1 , … , Y_{50} represent the concentrations of the 50 specimens taken at 250 cm. Then \bar{X}=88.5, \bar{Y}=110.6, s_X=49.4 and s_Y = 51.5. The sample sizes are n_X = n_Y =50. Both samples are large, so we can use expression (7.2). Since we want a 95% confidence interval, z_{\alpha / 2}=1.96 . The 95 % confidence interval for the difference \mu_Y-\mu_X is 110.6-88.5 \pm 1.96 \sqrt{49.4^2 / 50+51.5^2 / 50}\,\text{, or}\,22.1 \pm 19.8 \text{.}
\bar{X}-\bar{Y}\pm z_{\alpha / 2}\sqrt{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}} (7.2)