Chapter 10: Comparing Two Quantitative Variables Section 10.1: Scatterplots & Correlation

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1 Stat 300: Intro to Probability & Statistics Textbook: Introduction to Statistical Investigations Name: American River College Chapter 10: Comparing Two Quantitative Variables Section 10.1: Scatterplots & Correlation Now we will study relationships between two quantitative variables. A scatterplot is a graphical display of the relationship between two quantitative variables. o The explanatory variable goes on the horizontal (x-) axis, the response on the vertical (y-) axis. We examine a scatterplot for evidence of association between the variables. Three aspects of association to look for are: 1. Form a. The form of the association is linear if a straight line appears to summarize the relationship between the variables. 2. Direction a. Positive association means that larger values of one variable tend to appear with larger values of the other, and smaller values of one variable tend to appear with smaller values of the other. b. Negative association means that larger values of one variable tend to appear with smaller values of the other, and smaller values of one variable tend to appear with larger values of the other. 3. Strength a. Strength refers to the degree to which the data points follow a recognizable form. Example 1: House prices The following table reports the price and size (in square feet) for a sample of houses in Arroyo Grande, California. These data were obtained from the website zillow.com on February 7, 2007, for a random sample of houses listed on that site as recently sold. Address Price ($) Size (sq ft) Address Price ($) Size (sq ft) 2130 Beach St 311, Sycamore Dr 490, Lancaster Dr 344, Eman Ct 492, Golden West Pl 359, Adler St 500, Fair Oaks Ave 414, Cerro Vista Cir 510, Pearl Dr 459, Sycamore Dr 520, Rogers Ct 470, S Alpine St 541, Halcyon Rd 470, Woodland Ct 567, Poplar St 470, Paso Robles St 575, Fair Oaks Ave 474, Ocean St 580, Garfield Pl 475, Creekside Dr 625, a) What are the observational units here?

2 Stat 300 Text: Intro. to Statistical Investigations Section 10.1 Page 2 of 6 b) How many variables are reported in the table for each observational unit? What type (categorical or quantitative) is each variable? Consider the following scatterplot of price vs. size (our convention is to say y vs. x, with the first variable (y) on the vertical axis): c) Is house size positively or negatively associated with price? Would you describe the association as strong, moderate, or weak? Is the association roughly linear? 1. Direction: 2. Strength: 3. Form:

3 Stat 300 Text: Intro. to Statistical Investigations Section 10.1 Page 3 of 6 Example 2: Televisions and Life Expectancy The following table provides information on life expectancy and number of televisions per thousand people in a sample of 22 countries, as reported by the 2006 World Almanac and Book of Facts: Country Life Expectancy TVs per 1000 People Country Life Expectancy Angola Mexico Australia Morocco Cambodia Pakistan Canada Russia China South Africa Egypt Sri Lanka France Uganda Haiti United Kingdom Iraq United States Japan Vietnam Madagascar Yemen TVs per 1000 People a) Which of the countries listed has the fewest televisions per thousand people? Which has the most? What are those numbers? Fewest: Most: b) Enter the data (TVlive found in Data sets under Ch 10 or available in StatCrunch) into the Corr/Regression applet to produce a scatterplot of life expectancy vs. televisions per thousand people. Does there appear to be an association between the two variables? If so, describe its direction, strength, and form. c) Because the association is so strong, one might conclude that simply sending television sets to the countries with lower life expectancies would cause their inhabitants to live longer. Comment on this argument.

4 Stat 300 Text: Intro. to Statistical Investigations Section 10.1 Page 4 of 6 d) If two variables are strongly associated, does it follow that there must be a cause-andeffect relationship between them? Explain. e) In the case of life expectancy and television sets, suggest a confounding variable that is associated both with a country s life expectancy and with the prevalence of televisions in the country. Hint: Be sure to express this as a variable. Example 3: New Car Data On the last page, you will find nine scatterplots pertaining to variables measured on models of new cars in a) Arrange these plots from the most strongly negative to the most strongly positive association. Write the letter of the corresponding picture in the first row. Letter: R = Strong negative Moderate negative Virtually none Moderate positive Strong positive b) You will be told the value of the correlation coefficient (R) between the two variables for each scatterplot. Record these values below the letters in the table above. c) Based on the nine correlation values, what would you guess is the largest value that a correlation coefficient can have? How about the smallest?

5 Stat 300 Text: Intro. to Statistical Investigations Section 10.1 Page 5 of 6 d) Under what conditions would a correlation coefficient equal its largest possible value? Its smallest?e) How does the sign of the correlation relate to the direction of the association? f) How does the magnitude of the correlation relate to the strength of the association? g) Does order (which variable is x and which is y) matter when calculating a correlation coefficient? Explain. h) Is the correlation coefficient resistant to outliers? Explain how you can tell. Example 4: Correlation Guessing Game If you would like to see more examples of how the correlation coefficient is related to a scatterplot, please see the Correlation Guessing Game Applet.

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