In some studies, an investigator has n (x, y) pairs

Chapter 0, Problem 13.75

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In some studies, an investigator has n (x, y) pairs sampled from one population and m (x, y) pairs from a second population. Let b and denote the slopes of the first and second population lines, respectively, and let b and b denote the estimated slopes calculated from the first and second samples, respectively. The investigator may then wish to test the null hypothesis H0: b 0 (that is, b ) against an appropriate alternative hypothesis. Suppose that s2 , the variance about the population line, is the same for both populations. Then this common variance can be estimated by where SSResid and SSResid are the residual sums of squares for the first and second samples, respectively. With Sxx and denoting the quantity for the first and second samples, respectively, the test statistic is When H0 is true, this statistic has a t distribution based on (n m 4) df. The given data are a subset of the data in the article Diet and Foraging Model of Bufa marinus and Leptodactylus ocellatus (Journal of Herpetology [1984]: 138 146). The independent variable x is body length (cm) and the dependent variable y is mouth width (cm), with n 9 observations for one type of nocturnal frog and m 8 observations for a second type. Test at level .05 to determine if the slopes of the true regression lines for the two different frog populations are equal. (Summary statistics are given in the table.) Leptodactylus ocellatus x 3.8 4.0 4.9 7.1 8.1 8.5 8.9 9.1 9.8 y 1.0 1.2 1.7 2.0 2.7 2.5 2.4 2.9 3.2 Bufa marinus x 3.8 4.3 6.2 6.3 7.8 8.5 9.0 10.0 y 1.6 1.7 2.3 2.5 3.2 3.0 3.5 3.8 Sample size: 9 8 64.2 55.9 500.78 425.15 19.6 21.6 47.28 62.92 153.36 163.36