【正文】 % range. A)tt(P ,A ?? ??,At2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 856 D egr ee s of Fr ee do m1 3. 07 8 6. 31 4 12 . 70 6 31 . 82 1 63 . 65 72 1. 88 6 2. 92 4. 30 3 6. 96 5 9. 92 5. . . . . .. . . . . .10 1. 37 2 1. 81 2 2. 22 8 2. 76 4 3. 16 9. . . . . .. . . . . .200 1. 28 6 1. 65 3 1. 97 2 2. 34 5 2. 60 11. 28 2 1. 64 5 1. 96 2. 32 6 2. 57 6tA ? A = .05 A = .05 tA The t distribution is symmetrical around 0 = = ? The table provides the t values (tA) for which P(t? tA) = A Using the t Table 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 857 Student t Probabilities and Values ? Excel can calculate Student distribution probabilities and values. 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 858 ? The chisquared density function is given by: ? As before, the parameter ? is the number of degrees of freedom. Chi – Squared Distribution（ 卡方分配） ????2)(V)(E22??0χ f or e)(2 1)2/( 1)(f 22/1)2/(22/2 2 ?? ?? ??? ????2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 859 ? The chisquared distribution is not symmetrical ? The square, . , forces nonnegative values (. finding is illogical). ? Chi squared values can be found from the chi squared table, from Excel, or from Minitab. The Chi – Squared Distribution )0(P 2 ??2?2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 860 ? Table 5 in Appendix B makes it easy to lookup the right hand tail probability (A), for which The Chi – Squared Distribution A)(P 2A2 ?? ??2A?2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 861 ? For probabilities of this sort: ? We use 1–A, . we are determining The Chi – Squared Distribution A)(P 2 A12 ?? ???2 A1??2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 862 De g r e e s o ff r e e d o m1 0 . 0 0 0 0 3 9 3 0 . 0 0 0 1 5 7 1 . 3 . 8 4 1 4 6 6 . 6 3 4 9 7 . 8 7 9 4 4..10 2 . 1 5 5 8 5 2 . 5 5 8 2 1 . 1 8 . 3 0 7 2 3 . 2 0 9 3 2 5 . 1 8 8 2. . . . .. . . . . . . .0 5 10 15 20 25 30 35=.05 A =.99 Using the ChiSquared Table ? ? ? ? ? A ?2A To find ?2 for which P(?2??2)=.01, lookup the column labeled ? or ? ? 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 863 ? To find the point in a chisquared distribution with 8 degrees of freedom, such that the area to the right is .05, ? Look up the intersection of the 8 . row with the column, yielding a value of For Example 205.?2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 864 ? To find the point in a chisquared distribution with 8 degrees of freedom, such that the area to the left is .05, ? Look up the intersection of the 8 . row with the column, yielding a value of For Example 295.?2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 865 ChiSquared Probabilities and Values ? Excel can calculate chisquared distribution probabilities and values. 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 866 F Distribution（ F分配） ? The F density function is given by: ? Two parameters define this distribution, and like we’ve already seen these are again degrees of freedom. ? is the “numerator” degrees of freedom（ 分子的自由度） ? is the “denominator” degrees of freedom （ 分母的自由度） 0F f o r )F1(F)()2()2()2()F(f2212222121212111?????????????????????1?2?2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 867 ? 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 nonnegative) and is not symmetrical. F Distribution 2?2ν 2)F(E 222 ?????4ν )4()2()2(2)F(V222212122 ????????????2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 868 Determining Values of F ? The values of the F variable can be found in the F table, Excel, or from Minitab. ? The entries in the table are the values of the F variable of the right hand tail probability (A), for which A)FF(P A, 21 ????2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 869 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 lookup (Table 6) Numerator Degrees of Freedom : COLUMN Denominator Degrees of Freedom : ROW ,3,7 There are different tables for different values of A. Make sure you start with the correct table!! 7,3,05. ?2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 870 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! 1221,A,A1F1F??????2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 871 F Probabilities and Values ? Excel can calculate F distribution probabilities and values.