9 Simple Linear Regression

Where We’ve Been

  • Presented methods for estimating and testing population parameters (e.g., the mean, proportion, and variance) for a single sample

  • Extended these methods to allow for a comparison of population parameters for multiple samples

Where We’re Going

  • Introduce the straight-line (simple linear regression) model as a means of relating one quantitative variable to another quantitative variable (9.1)

  • Assess how well the simple linear regression model fits the sample data (9.29.4)

  • Introduce the correlation coefficient as a means of relating one quantitative variable to another quantitative variable (9.5)

  • Utilize the simple linear regression model to predict the value of one variable from a specified value of another variable (9.6, 9.7)

  • Introduce a nonparametric test for correlation (9.8)

Statistics in Action Can “Dowsers” Really Detect Water?

The act of searching for and finding underground supplies of water with the use of nothing more than a divining rod is commonly known as “dowsing.” Although widely regarded among scientists as no more than a superstitious relic from medieval times, dowsing remains popular in folklore, and to this day, there are individuals who claim to have this mysterious skill. In fact, as recently as March 2014, a group of California farmers hired dowsers to find water during a drought.

Many dowsers claim that they respond to “earthrays” that emanate from the water source. Earthrays, say the dowsers, are a subtle form of radiation that is potentially hazardous to human health. As a result of these claims, in the mid-1980s the German government conducted a two-year experiment to investigate the possibility that dowsing is a genuine skill. If such a skill could be demonstrated, reasoned government officials, then dangerous levels of radiation in Germany could be detected, avoided, and disposed of.

A group of university physicists in Munich, Germany, was provided a grant of 400,000 marks (about $250,000) to conduct the study. Approximately 500 candidate dowsers were recruited to participate in preliminary tests of their skill. To avoid fraudulent claims, the 43 individuals who seemed to be the most successful in the preliminary tests were selected for the final, carefully controlled, experiment.

The researchers set up a 10-meter-long line on the ground floor of a vacant barn, along which a small wagon could be moved. Attached to the wagon was a short length of pipe, perpendicular to the test line, that was connected by hoses to a pump with running water. The location of the pipe along the line for each trial of the experiment was assigned by a computer-generated random number. On the upper floor of the barn, directly above the experimental line, a 10-meter test line was painted. In each trial, a dowser was admitted to this upper level and required, with his/her rod, stick, or other tool of choice, to ascertain where the pipe with running water on the ground floor was located.

Each dowser participated in at least one test series constituting a sequence of from 5 to 15 trials (typically, 10), with the pipe randomly repositioned after each trial. (Some dowsers undertook only 1 test series, whereas selected others underwent more than 10 test series.) Over the two-year experimental period, the 43 dowsers participated in a total of 843 tests. The experiment was “double blind” in that neither the observer (researcher) on the top floor nor the dowser knew the pipe’s location, even after a guess was made. [Note: Before the experiment began, a professional magician inspected the entire arrangement for potential deception or cheating by the dowsers.]

For each trial, two variables were recorded: the actual location of the pipe (in decimeters from the beginning of the line) and the dowser’s guess (also measured in decimeters). On the basis of an examination of these data, the German physicists concluded in their final report that although most dowsers did not do particularly well in the experiments, “some few dowsers, in particular tests, showed an extraordinarily high rate of success, which can scarcely if at all be explained as due to chance…a real core of dowser-phenomena can be regarded as empirically proven . . .” (Wagner, Betz, and König, 1990. Final Report 01 KB8602, Federal Ministry for Research and Technology).

This conclusion was critically assessed by Professor J. T. Enright of the University of California at San Diego (Skeptical Inquirer, Jan./Feb. 1999). In the Statistics in Action Revisited sections of this chapter, we demonstrate how Enright concluded the exact opposite of the German physi­cists. [Note: Enright’s assessment of the German study led other researchers, notably Vogtand Hyman (2000), Carroll (2003), and Whitaker (2006), to conduct their own scientific studies debanking water dowsing.]

Statistics in Action Revisited

  • Estimating a Straight-Line Regression Model for the Dowsing Data (p. 510)

  • Assessing How Well the Straight-Line Model Fits the Dowsing Data (p. 527)

  • Using the Coefficients of Correlation and Determination to Assess the Dowsing Data (p. 538)

  • Using the Straight-Line Model to Predict Pipe Location for the Dowsing Data (p. 546)

In Chapters 57, we described methods for making inferences about population means. The mean of a population has been treated as a constant, and we have shown how to use sample data to estimate or to test hypotheses about this constant mean. In many applications, the mean of a population is not viewed as a constant, but rather as a variable. For example, the mean sale price of residences in a large city might be treated as a variable that depends on the number of square feet of living space in the residence. The relationship might be

Meansaleprice=$30,000+$60 (Squarefeet)

This formula implies that the mean sale price of 1,000-square-foot homes is $90,000, the mean sale price of 2,000-square-foot homes is $150,000, and the mean sale price of 3,000-square-foot homes is $210,000.

In this chapter, we discuss situations in which the mean of the population is treated as a variable, dependent on the value of another variable. The dependence of the residential sale price on the number of square feet of living space is one illustration. Other examples include the dependence of the mean reaction time on the amount of a drug in the bloodstream, the dependence of the mean starting salary of a college graduate on the student’s GPA, and the dependence of the mean number of years to which a criminal is sentenced on the number of previous convictions.

Here, we present the simplest of all models relating a populating mean to another variable: the straight-line model. We show how to use the sample data to estimate the straight-line relationship between the mean value of one quantitative variable, y, as it relates to a second quantitative variable, x. The methodology of estimating and using a straight-line relationship is referred to as simple linear regression analysis.

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