inference for the difference in means -two normal distributions and variance unknown

Horng-Chyi HorngHorng-Chyi Horng Statistics II_FiveStatistics II_Five 11

Inference for the Difference in MeansInference for the Difference in Means-Two Normal Distributions and-Two Normal Distributions and Variance UnknownVariance Unknown

&5-3 (&9-3)

Why? Why?


2

11

2

,

.2

1

21

222

211

21

1

222

1

211

2

2

2222

2111

21

nn

SnSn

nn

XXXX

S

is for σestimationtheTherefore

and SXion with nd populatrom samples fand n

, and SXion with st populatrom samples fWe have n

n

jj

n

ii

p


Replacing by Sp, we have

.

11 ,

21

2121222

21

nn

XXZthereforeSince


Example 9-5Example 9-5

Two catalysts are being analyzed to determine Two catalysts are being analyzed to determine how they affect the mean yield of a chemical how they affect the mean yield of a chemical process. Specially, catalyst 1 is currently in use, process. Specially, catalyst 1 is currently in use, but catalyst 2 is acceptable. Since catalyst 2 is but catalyst 2 is acceptable. Since catalyst 2 is cheaper, it should be adopted, providing it does cheaper, it should be adopted, providing it does not change the process yield. A test is run in the not change the process yield. A test is run in the pilot plant and results in the data shown in Table pilot plant and results in the data shown in Table 9-2. Is there any difference between the mean 9-2. Is there any difference between the mean yields? Use yields? Use = 0.05 and assume equal = 0.05 and assume equal variances.variances.


Table 9-2 Catalyst Yield Data, Example 9-5Table 9-2 Catalyst Yield Data, Example 9-5


have, we still and S with S and σplacing σ 22

21

22

21Re

is distributed approximately as t with degrees of freedom given by


C.I. on the Difference in MeansC.I. on the Difference in Means


Paired t-TestPaired t-Test

When the observations on the two populations of interest aWhen the observations on the two populations of interest are collected in re collected in pairspairs..

Let (XLet (X1111, X, X2121), (X), (X1212, X, X2222), …, (X), …, (X1n1n, X, X2n2n) be a set of n paired o) be a set of n paired o

bservations, in which Xbservations, in which X1j1j~(~(11, , 1122) and X) and X2j2j~(~(22, , 22

22) and D) and Djj

= X= X1j1j – X – X2j2j, j = 1, 2, …, n. Then, to test H, j = 1, 2, …, n. Then, to test H00: : 11= = 22 is the s is the s

ame as performing a one-sample t-test Hame as performing a one-sample t-test H00: : D D = 0 since= 0 since

D D = E(X= E(X11-X-X22) = E(X) = E(X11)-E(X)-E(X22) = ) = 1 1 - - 22

&5-4 (&9-4)


Example 9-9 An article in the Journal of Strain Analysis Example 9-9 An article in the Journal of Strain Analysis compares several methods for predicting the shear strength for steel compares several methods for predicting the shear strength for steel plate girders. Data for two of these methods are shown in Table plate girders. Data for two of these methods are shown in Table below. We wish to determine whether there is any difference between below. We wish to determine whether there is any difference between the two methods.the two methods.


Paired Vs. UnpairedPaired Vs. Unpaired

• If the experimental units are relatively homogeneous (small ) and the correlation within pairs is small, the gain in precision attributable to pairing will be offset by the loss of degrees of freedom, so an independent-sample experiment should be used.

• If the experimental units are relatively heterogeneous (large ) and there is large positive correlation within pairs, the paired experiment should be used. Typically, this case occurs when the experimental units are the same for both treatments; as in Example 9-9, the same girders were used to test the two methods.


A Confidence Interval for A Confidence Interval for DD


Table 9-5 Table 9-5 Time in Seconds to Parallel Park Two AutomobilesTime in Seconds to Parallel Park Two Automobiles

inference for the difference in means -two normal distributions and variance unknown

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