502 CHAPTER 9 Correlation and Regression Prediction Intervals Recall from Section 9.1 that one of the requirements for calculating a correlation coefficient is that the two variables x and y have a bivariate normal distribution. Two variables have a bivariate normal distribution when for any fixed values of x the corresponding values of y are normally distributed, and for any fixed values of y the corresponding values of x are normally distributed. x xn x1 y1 x2 y = mx + b y2 yn ∧ ∧ ∧ ∧ y Bivariate Normal Distribution Because regression equations are determined using random samples of paired data and because x and y are assumed to have a bivariate normal distribution, you can construct a prediction interval for the true value of y. To construct the prediction interval, use a t@distribution with n - 2 degrees of freedom. Given a linear regression equation ny = mx + b and x 0, a specific value of x, a c-prediction interval for y is ny - E 6 y 6 ny + E where E = tc seC1 + 1 n + n1x0 - x2 2 nΣx2 - 1Σx22 . The point estimate is ny and the margin of error is E. The probability that the prediction interval contains y is c (the level of confidence), assuming that the estimation process is repeated a large number of times. DEFINITION Constructing a Prediction Interval for y for a Specific Value of x In Words In Symbols 1. Identify the number n of pairs of d.f. = n - 2 data and the degrees of freedom. 2. Use the regression equation nyi = mxi + b and the given x@value to find the point estimate ny. 3. Find the critical value tc that Use Table 5 in Appendix B. corresponds to the given level of confidence c. 4. Find the standard error of se = CΣ1yi - nyi2 2 n - 2 estimate se. 5. Find the margin of error E. E = tc seC1 + 1 n + n1x0 - x2 2 nΣx2 - 1Σx22 6. Find the left and right Left endpoint: ny - E endpoints and form the Right endpoint: ny + E prediction interval. Interval: ny - E 6 y 6 ny + E GUIDELINES Study Tip The formulas for se and E use the quantities Σ1yi - nyi2 2, 1Σx22, and Σx2. Use a table to calculate these quantities.
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