21 7 79 1 1 21 1 14 68 12 771 1 21 1 D D DD D n s t d n st d Notice

# 21 7 79 1 1 21 1 14 68 12 771 1 21 1 d d dd d n s t d

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21 . 7 79 . 4 14 / 68 . 12 771 . 1 21 . 1 14 / 68 . 12 771 . 1 21 . 1 / /   D D D D D n s t d n s t d Notice that the confidence interval on µ D includes zero. This implies that, at the 90% level of confidence, the data do not support the claim that the two cars have different mean parking times µ 1 and µ 2 . That is, the value µ D = µ 1 - µ 2 = 0 is not inconsistent with the observed data. Subject 1( x 1 j ) 2( x 2 j ) ( d j ) 1 37.0 17.8 19.2 2 25.8 20.2 5.6 3 16.2 16.8 -0.6 4 24.2 41.4 -17.2 5 22.0 21.4 0.6 6 33.4 38.4 -5.0 7 23.8 16.8 7.0 8 58.2 32.2 26.0 9 33.6 27.8 5.8 10 24.4 23.2 1.2 11 23.4 29.6 -6.2 12 21.2 20.6 0.6 13 36.2 32.2 4.0 14 29.8 53.8 -24.0 Table 10-4 The F Distribution Inferences on the Variances of Two Normal Populations 14 Let W and Y be independent chi-square random variables with u and v degrees of freedom respectively. Then the ratio (10-28) has the probability density function (10-29) and is said to follow the distribution with u degrees of freedom in the numerator and v degrees of freedom in the denominator. It is usually abbreviated as F u,v . v Y u W F / / x x v u v u x v u v u x f v u u u 0 , 1 2 2 2 ) ( /2 1 /2) ( /2 Sec 10-5 Inferences on the Variances of Two Normal Populations Sec 10-5 Inferences on the Variances of Two Normal Populations 15 Let be a random sample from a normal population with mean µ 1 and variance , and let be a random sample from a second normal population with mean µ 2 and variance . Assume that both normal populations are independent. Let and be the sample variances. Then the ratio has an F distribution with n 1 1 numerator degrees of freedom and n 2 1 denominator degrees of freedom. 1 1 12 11 , , , n X X X 2 1 2 2 22 21 , , , n X X X 2 2 2 1 S 2 2 S 2 2 2 2 2 1 2 1 / / S S F The F Distribution Confidence Interval on the Ratio of Two Variances If and are the sample variances of random samples of sizes n 1 and n 2 , respectively, from two independent normal populations with unknown variances and , then a 100(1   )% confidence interval on the ratio is (10-33) where and are the upper and lower /2 percentage points of the F distribution with n 2 1 numerator and n 1 1 denominator degrees of freedom, respectively. A confidence interval on the ratio of the standard deviations can be obtained by taking square roots in Equation 10-33. 2 1 s 2 2 s 2 1 2 2 2 2 2 1 / 1 , 1 , /2 2 2 2 1 2 2 2 1 1 , 1 /2, 1 2 2 2 1 1 2 1 2 n n n n f s s f s s 1 , 1 , /2 1 2 n n f 2 1 /2, 1, 1 n n f  Example 10-15 Surface Finish for Titanium Alloy A company manufactures impellers for use in jet-turbine engines. One of the operations involves grinding a particular surface finish on a titanium alloy component. Two different grinding processes can be used, and both processes can produce parts at identical mean surface roughness. The manufacturing  #### You've reached the end of your free preview.

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