# Distribution is given by e f w w 2 it depends only on

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distribution is given by E ( F ) = w /( w 2 ) ; it depends only on the second degrees of freedom and is slightly larger than 1. The variance depends on both v and w . Table D (at the end of the book) gives selected percentiles. For example, the 95 and 99th percentiles of an F (5,8) distribution are given by 3.69 and 6.63, respectively. Suppose U χ 2 v and V χ 2 w , with U and V independent. Then F = U / v V / w has an F distribution with v and w degrees of freedom. A comment on statistical tables : Most computer programs calculate cumu- lative probabilities and percentiles (the inverse of the cumulative probabilities) for a wide selection of distributions. For some programs (such as EXCEL) the calculation of percentiles requires the speci fi cation of the upper tail area. EXERCISES 2.1. Determine the 95 and 99th percentiles of a. The normal distribution with mean 10 and standard deviation 3; b. The t distributions with 10 and 25 degrees of freedom; c. The chi-square distributions with 1, 4, and 10 degrees of freedom; d. The F distributions with 2 and 10, and 4 and 10 degrees of freedom. 2.2. It is a fact that two distributions are the same if (all) their percentiles are identical. a. Convince yourself, by looking up several percentiles, that the square of a standard normal distribution is the same as a chi-square distribution with one degree of freedom. Determine the percentile of the χ 2 1 and the percentile of the square of a standard normal distribution, Z 2.3. For each of the four sets of data given below (see Anscombe, 1973), plot y versus x . The data are given in the fi le anscombe . Fit a straight line model to each of the data sets giving least squares estimates, ANOVA table, and R 2 . Compute the correlation coef fi cient between y and x for each data set. Comment on your results. Would a linear regression of y on x be appropriate in all cases? Discuss. Set 1 Set 2 Set 3 Set 4 x y x y x y x y 2 , and show that they are the same. Use the fact that P ( Z 2 z ) = P ( z Z z ) . Hence, for example, the 95th percentile of Z 2 is the same as the 97 . 5th percentile of Z . 4 4.26 4 3.10 4 5.39 8 6.58 5 5.68 5 4.74 5 5.73 8 5.76 6 7.24 6 6.13 6 6.08 8 7.71 7 4.82 7 7.26 7 6.42 8 8.84 8 6.95 8 8.14 8 6.77 8 8.47 9 8.81 9 8.77 9 7.11 8 7.04 10 8.04 10 9.14 10 7.46 8 5.25 11 8.33 11 9.26 11 7.81 8 5.56 12 10.84 12 9.13 12 8.15 8 7.91 13 7.58 13 8.74 13 12.74 8 6.89 14 9.96 14 8.10 14 8.84 19 12.50
b. Convince yourself, by looking up several percentiles, that the square of a t distribution with v degrees of freedom is the same as the F ( 1 , v ) distribution. 2.4. A car dealer is interested in modeling the relationship between the weekly number of cars sold and the daily average number of
Abraham Abraham ˙ C02 November 8, 2004 0:36 Exercises 57 salespeople who work on the showroom fl oor during that week. The dealer believes that the relationship between the two variables can be described by a straight line. The following data were supplied by the car dealer: Average. No.