Keller: Stats for Mgmt & Econ, 7th Ed July 17, 2006

Transcription

1 Chapter 8 Other Continuous Distributions 8.1 Other Continuous Distributions Three other important continuous distributions which will be used extensively in later sections are introduced here: Student t Distribution, Chi-Squared Distribution, and F Distribution. 8.2 Student t Distribution Here the letter t is used to represent the random variable, hence the name. The density function for the Student t distribution is as follows (nu) is called the degrees of freedom, and (Gamma function) is (k)=(k-1)(k-2) (2)(1) 8.3 of Thomson Learning, Inc. 1

2 Student t Distribution Much like the standard normal distribution, the Student t distribution is mound shaped and symmetrical about its mean of zero: The mean and variance of a Student t random variable are E(t) = 0 and V(t) = for > Student t Distribution In much the same way that and define the normal distribution,, the degrees of freedom, defines the Student t Distribution: Figure 8.24 As the number of degrees of freedom increases, the t distribution approaches the standard normal distribution. 8.5 Determining Student t Values The student t distribution is used extensively in statistical inference. Table 4 in Appendix B lists values of That is, values of a Student t random variable with degrees of freedom such that: The values for A are pre-determined critical values, typically in the 10%, 5%, 2.5%, 1% and 1/2% range. 8.6 of Thomson Learning, Inc. 2

3 Using the table (Table 4) for values For example, if we want the value of t with 10 degrees of freedom such that the area under the Student t curve is.05: Area under the curve value (t A ) : COLUMN t.05,10 t.05,10 =1.812 Degrees of Freedom : ROW 8.7 Student t Probabilities and Values Excel can calculate Student distribution probabilities and values. 8.8 Chi-Squared Distribution The chi-squared density function is given by: As before, the parameter freedom. is the number of degrees of Figure of Thomson Learning, Inc. 3

4 Chi-Squared Distribution Notes: The chi-squared distribution is not symmetrical The square, i.e., forces non-negative values (e.g. finding P( < 0) is illogical). Table 5 in Appendix B makes it easy to look-up probabilities of this sort, i.e. P( > ) = A: 8.10 Chi-Squared Distribution For probabilities of this sort: We use 1 A, i.e. we are determining P( < ) = A 8.11 For Example To find the point in a chi-squared distribution with 8 degrees of freedom, such that the area to the right is.05, Look up the intersection of the 8 d.f. row with the column, yielding a value of of Thomson Learning, Inc. 4

5 For Example To find the point in a chi-squared distribution with 8 degrees of freedom, such that the area to the left is.05, Look up the intersection of the 8 d.f. row with the column, yielding a value of For Example This makes sense: =2.73 =15.51 Remember the axis starts and zero and increases! 8.14 Chi-Squared Probabilities and Values Excel can calculate chi-squared distribution probabilities and values of Thomson Learning, Inc. 5

6 F Distribution The F density function is given by: F > 0. Two parameters define this distribution, and like we ve already seen these are again degrees of freedom. is the numerator degrees of freedom and is the denominator degrees of freedom F Distribution The mean and variance of an F random variable are given by: and The F distribution is similar to the distribution in that its starts at zero (is non-negative) and is not symmetrical Determining Values of F For example, what is the value of F for 5% of the area under the right hand tail of the curve, with a numerator degree of freedom of 3 and a denominator degree of freedom of 7? Solution: use the F look-up (Table 6) There are different tables for different values of A. Make sure you start with the correct table!! F.05,3,7 F.05,3,7 =4.35 Denominator Degrees of Freedom : ROW Numerator Degrees of Freedom : COLUMN 8.18 of Thomson Learning, Inc. 6

7 Determining Values of F For areas under the curve on the left hand side of the curve, we can leverage the following relationship: Pay close attention to the order of the terms! 8.19 F Probabilities and Values Excel can calculate F distribution probabilities and values of Thomson Learning, Inc. 7