Exam 4: Describing Bivariate Numerical Data

What is it that the correlation coefficient measures?

The least squares line passes through the point ( xy, ) .

In the 19th Century, venereal diseases were the major preventable diseases strikingsoldiers far from home. During the American Civil War, the United States Army keptrecords on soldiers diagnosed with syphilis and gonorrhea. An analysis of theincidence of these diseases for a sample of months is presented below. (Incidencerate is the number of cases per 100,000 soldiers.)Summary statistics from Regression AnalysisIncidence of syphilis and gonorrhea forthe years 1861 - 1865 a) For a month with an incidence rate for Syphilis that is one standard deviation abovethe mean rate, what would be the predicted incidence rate for gonorrhea? b) What proportion of the variability in the gonorrhea incidence rate is accounted for bysyphilis incidence rate? c) Suppose you are asked to assess the accuracy of the prediction of the gonorrheaincidence rate. What information above could be used to answer this question?

The coefficient of determination is equal to the positive square root of thecorrelation coefficient, r.

Generally, a relatively small value of r² is associated with a relativelysmall se .

As early as 3 years of age, children begin to show preferences for playing withmembers of their own sex, and report having more same-sex than opposite-sexfriends. Researchers believe that this may be the result of perceived differences inpersonality. In a study of 3rd and 4th graders' views on a number personality traits,children were asked to rate on a "5-point" scale:-2 = "someone possessing that trait is probably a boy"-1 = "someone possessing that trait might be a boy"0 = "can't tell"1 = "someone possessing that trait might be a girl"2 = "someone possessing that trait is probably a girl"A scatterplot of the data is presented below. A single point represents the (averagegirls' rating, average boys' rating) for a given trait. a) Circle the single point which represents the most influential observation. Whataspect of this point makes it the most influential? b) Suppose a personality trait similar to those used in the survey was given anaverage of 0.0 ("can't tell") by the girls. The predicted boys' average rating wouldbe closest to which of the 5 categories described above? c) The traits plotted above are those the researchers believe are "positive" traits, suchas "mature," "honest," and "polite." The researchers thought that on average girlswould rate these positive traits as characteristic of girls to a greater extent thanboys would. What aspects of the plot and/or regression analysis presented aboveare consistent with this thinking?

What is it that the correlation coefficient measures?

One of the properties of correlation coefficient, r, is: "The value of r does not dependon the unit of measurement for either variable." In your own words, what does thismean?

The data below were gathered on a random sample of 7 male banded black-footedalbatrosses of known age. In an effort to monitor diseases of these animals, biologistswould like to be able to estimate the lifespan of healthy albatrosses in the largerpopulation. In males of this species gonad size (the size of the sex gland) isassociated with age. a) What is the value of the correlation coefficient for these data? b) What is the equation of the least squares line describing the relationship betweenx = Gonad Size and y = Age.c) If these albatrosses are representative of the population of all albatrosses, whatwould you predict to be the age of a male albatross with a gonad size of 50 sq.mm? Show any calculations below.d) The largest albatross gonad size in the sample, 96 sq mm, is associated with anage of 23.64 years. These animals are thought to live for up to 40 years. Would itbe reasonable to use the equation from part (b) above to predict the age for analbatross with a gonad size of 150 sq mm? Why or why not?