In New Hampshire, about half the counties mandate the use of...

In New Hampshire, about half the counties mandate the use of reformulated gasoline. This has led to an increase in the contamination of groundwater with methyl tert-butyl ether (MTBE). Environmental Science \& Technology (Jan. 2005) reported on the factors related to MTBE contamination in public and private New Hamsphire wells. Data were collected on a sample of 223 wells. These data are saved in the MTBE file. Three of the variables are qualitative in nature: well class (public or private), aquifer (bedrock or unconsolidated), and detectable level of MTBE (below limit or detect). [Note: A detectable level of $\mathrm{MTBE}$ occurs if the MTBE value exceeds .2 microgram per liter.] The data on 11 selected wells are shown in the accompanying table. a. Use graphical methods to describe each of the three qualitative variables for all 223 wells. b. Use side-by-side bar charts to compare the proportions of contaminated wells for private and public well classes. c. Use side-by-side bar charts to compare the proportions of contaminated wells for bedrock and unconsolidated aquifiers. d. What inferences can be made from the bar charts of parts $a-c$ ?

In New Hampshire, about half the counties mandate the use of reformulated gasoline. This has led to an increase in the contamination of groundwater with methyl tert-butyl ether (MTBE). Environmental Science \& Technology (Jan. 2005) reported on the factors related to MTBE contamination in public and private New Hamsphire wells. Data were collected on a sample of 223 wells. These data are saved in the MTBE file. Three of the variables are qualitative in nature: well class (public or private), aquifer (bedrock or unconsolidated), and detectable level of MTBE (below limit or detect). [Note: A detectable level of $\mathrm{MTBE}$ occurs if the MTBE value exceeds .2 microgram per liter.] The data on 11 selected wells are shown in the accompanying table.
a. Use graphical methods to describe each of the three qualitative variables for all 223 wells.
b. Use side-by-side bar charts to compare the proportions of contaminated wells for private and public well classes.
c. Use side-by-side bar charts to compare the proportions of contaminated wells for bedrock and unconsolidated aquifiers.
d. What inferences can be made from the bar charts of parts $a-c$ ?

Key Concepts

Inference from Graphical Data

Inference from graphical data involves interpreting visual representations to draw conclusions about the relationships or differences among variables. By analyzing features such as the height and grouping of bars in charts, one can assess trends, evaluate comparisons, and make informed decisions about the underlying factors driving observed patterns in the data.

Proportional Analysis

Proportional analysis involves calculating and comparing the percentages or relative frequencies of categories within the data. This approach is crucial when studying contamination rates, as it allows researchers to discern whether certain groups (like different well types) exhibit significantly higher or lower proportions of contamination.

Graphical Methods for Categorical Data

Graphical methods, such as bar charts, are used to represent categorical data visually. These methods help in summarizing and presenting the distribution of qualitative variables in a clear and concise manner, making it easier to observe patterns, frequencies, and anomalies in the data.

Qualitative Variables

Qualitative variables represent categorical data that classifies observations into distinct groups based on non-numeric attributes. In contexts like environmental studies, they include classifications such as well type or contamination status, and are typically summarized by counts or proportions rather than means or standard deviations.

Side-by-Side Bar Charts

Side-by-side bar charts are a specific type of bar graph that allows for the comparison of two or more groups across the same categorical variable. They place related bars next to each other to directly compare proportions or frequencies between different groups, facilitating insights into differences in characteristics such as contamination rates across well classes or aquifer types.

Transcript

00:01 For this problem, we are looking at essentially a sort of environmental study about contamination of groundwater with methyl -tertutal ether, or mtbe.

00:11 We have three different qualitative variables, the well class being public or private, the aquifer, bedrock or unconsolidated, and whether or not something exceeds the detectable level of mtbe.

00:24 So for part a, we are asked to use graphical methods to describe each of the qualitative variables.

00:30 For all 223 wells.

00:33 One moment here.

00:35 So i'll note here that i don't have access to the actual data file that is intended to go along with the problem, but i will show you an example here.

00:43 So the first thing that'll do is create a frequency table for well class.

00:49 We have two types, private and public.

00:54 We can see that there are only two private and the rest of the 11, so 9, will be public.

00:58 So one graphical method for this, which would be quite relevant, would be a pie chart.

01:06 I'll say, well class frequency.

01:11 Grab that data, go to insert charts, pie chart.

01:17 Then we'll look at the aquifer type frequency.

01:25 Thankfully only two categories.

01:27 So there's bedrock and unconsolidated, abbreviate as uncons.

01:34 So there are four bedrock, and then there are four bedrock, and then, the remainder of the 11.

01:39 Oh, 11 minus 4, that would be 7.

01:42 So there would be 7 remaining, would be unconsolidated.