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

This preview shows page 13 - 18 out of 25 pages.

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
Image of page 13
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
Image of page 14
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
Image of page 15
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 
Image of page 16
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
Image of page 17
Image of page 18

You've reached the end of your free preview.

Want to read all 25 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes