Question

1 - A concerned group of citizens has complained to Environmental Health Services (EHS) that the water in a nearby river is contaminated. They are accusing a local industrial company of polluting the river and are calling for the company to be shut down or to lessen their emissions. The EHS decides to run a pH-level test on the water to determine if it is, in fact, polluted. The EHS has stated that the pH-level of natural water in the United States should fall between 6.5 and 8.5, with 7.0 being neutral. They hire your firm to do the analysis. You randomly select 30 water samples and measure their individual pH-levels. The results of your study are listed in the table below:

6.78 6.69 6.84 6.79 6.92

6.70 6.71 6.93 6.29 6.92

6.85 6.61 6.56 6.62 6.77

6.33 7.00 7.10 6.61 6.26

6.58 6.43 7.34 6.51 6.81

7.01 6.37 6.61 7.07 6.58

a) Provide the client with a graphical display of the sample data you collected (this could be a stem and leaf plot, a box and whisker plot, or a histogram).

b) Calculate the mean, median, and standard deviation of your samples.

c) Explain each of the findings from Question b for your client.

Answer 1

The following is the R code to plot stem and leaf plot, box and whisker plot and histogram; and also to calculate mean, median and standard deviation of the sample.

+ + + > pH<-c(6.78, 6.69, 6.84, 6.79, 6.92, 6.70 6.71, 6.936.29, 6.92, 6.85, 6.61, 6.56, 6.62, 6.77, 6.33, 7.00 7.10, 6.61, 6.26, 6.58, 6.43, 7.34, 6.51, 6.81, 7.01, 6.37, 6.61, 7.07, 6.58) > 3tem (pH) + + The decimal point is i digit (3) to the left of the 62 | 6937 64 31688 66 | 1112901789 68 | 145223 70 | 0170 724 > boxplot (pH, main = "Box and whisker plot", col="cyan"), > hist (pH, main = "Histogram",xlab = "pH Level3", col="cyan") > mean (pH);median (pH); var (pH): 3d (pH) [1] 6.719667 [1] 6.705 [1] 0.06552057 [1] 0.2559699

a) Stem and Leaf Plot

stem (pH) The decimal point is i digit(s) to the left of the l 62 | 6937 64 31688 66 | 1112901789 68 | 145223 70 | 0170 72 | 4

Box and whisker plot

6.4 6.8 7.2 Box and whisker plot

Histogram

Histogram 10 8 6 Frequency 4 2 0 6.2 6.4 6.6 6.8 70 72 7.4 pH Levels

b) Mean = 6.719667
Median = 6.705
Variance = 0.06552057
Standard deviation = 0.2559699

c) The median and the mean both measure central tendency. The standard deviation is a summary measure of the differences of each observation from the mean. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation.

From part (b) we have that the mean and median of the sample of pH levels is approximately 6.7 and this lies between 6.5 and 8.5. The standard deviation of the sample is calculated to be 0.2559 which is very small. Since we know that a low standard deviation indicates that the data points tends to be close to the mean of the sample with less variation we can say that the pH levels of the river water is concentrated around 6.719.

So according to EHS's statement i.e; the pH-level of natural water in the United States should fall between 6.5 and 8.5, with 7.0 being neutral we can conclude that water in the nearby river is not contaminated.