Test of Hypotheses: Two Sample

Test of Hypotheses: Two Sample.Outlines: Inference on the difference in means of two

normal distributions, variance known Inference on the difference in means of two

normal distributions, variance unknown Paired t-test Inference on the variances of two normal

distributions Inference on the two population proportions

Hypothesis testing

Engineers and scientist are often interested in comparing two difference conditions to determine whether either conditions produce a significant effect on the response.

Condition => Treatment Cause and effect relationship: the difference in

treatments resulted in the difference in response.

Case I

Inference on the difference in means of two normal distributions, variance known

Hypothesis

Test Statistic

We should reject H0 if

0211

0210

:

:

H

H

2

22

1

21

0210

nn

XXZ

2/02/0 ZZorZZ

Case I

Ex. A product developer is interested in reducing the drying time of a primer paint. Two formulations (old, new) of the paint are tested. The sd of drying time is 8 mins. Ten specimens are paint with formulation 1, and another 10 specimens are painted with formulation 2; the 20 specimens are paint in random order. The two sample average drying times are

What conclusion can be drawn about the effectiveness of the new ingredient, using =0.05

1. Parameter of interest is the difference in mean drying time,

2.

3. =0.05

4. Test statistic ,

5. reject H0 if

6. Calculate Z0 = 2.52

7. Conclusion : since , we reject H0 at 0.05 significance level. Adding new ingredient to the paint significantly reduces the drying time.

112,121 21 xx

645.12/0 ZZ

21

211210 :,0: HH

2

22

1

21

0210

nn

XXZ

112,121,10,8,0 2121210 xxnn

2/0 ZZ

Case I

Sample size Using Operating Characteristic Curve (OC Curve)

If the value of , we can use the formula to calculate the value of n1 when n2 is fixed,

22

21

0

22

21

021 ||||

d n=n1=n2

21 nn

2221

21

22

21

// nnn

Case I

Sample Size formulas

Case I

Confidence Interval

The error in estimating µ1-µ2 by will be less than E at 100(1- )% confidence. Therequired sample size from each population is

21 xx

Case I

Ex. Tensile strength tests were performed on two different grades of aluminum spars. From past experience with the spar manufacturing process and testing procedure, the standard deviations of the tensile strengths are assumed to be known. The data obtained are as follows:

Find a 90% confidence interval on the difference in mean strength µ1-µ2

5.1,5.74,12,1,6.87,10 222111 xnxn

Case I.I

We can use the concept of Case I for the cases that we don’t know exactly about the population distribution (may be not normal distribution) and the number of sample size are large. (n1, n2 >=40)

Inference on the difference in means of two normal distributions, variance unknown

Case 1: Hypothesis

Test Statistic


Case II.I

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0210

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:

H

H

2

)1()1(,

11 21

222

2112

21

0210

nn

SnSnS

nnS

XXT p

p

2,2/02,2/0 2121 nnnn ttortt

Pooled Estimator of variance

22

21

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

Case II.I

Case II.I

1. Parameter of interest: µ1 and µ2, the mean process yield using C1, and C2

2. H0: µ1-µ2=0 or H0: µ1=µ2 , H1: µ1≠µ2

3. =0.05

4. Test statistic is

5. Reject H0 if

6. Calculate t0;

7. Conclusion H0 cannot be rejected. At the 0.05 level of significant, we do not have a strong evidence to conclude that C2 results in a mean yield differ from C1

2

)1()1(,

11 21

222

2112

21

0210

nn

SnSnS

nnS

XXT p

p

145.2145.2 14,025.0014,025.00 ttortt

Inference on the difference in means of two normal distributions, variance unknown

Case 2: Hypothesis

Test Statistic


Case II.II

0211

0210

:

:

H

H

2

2

1

1

021*0

nS

nS

XXT

,2/0,2/0 ttortt

Degree of freedom

22

21

Case II.II

Ex. Arsenic concentration in public drinking water supplies is a potential health risk. An article in the Arizona Republic reported drinking water arsenic concentration in parts per billion (ppb) for 10 metropolitan Phoenix communities and 10 communities in rural Arizona.

Case II.II

22

21

Case II.II

1. Parameter of interest: µ1 and µ2, the mean arsenic concentration of two regions

2. H0: µ1-µ2=0 or H0: µ1=µ2 , H1: µ1≠µ2

3. =0.05

4. Test statistic:

5. We should reject H0 if

6. Compute t0

2

2

1

1

021*0

nS

nS

XXT

,025.00,025.00 ttortt

Case II.II

7. Conclusion: t0<t0.025,13, we reject H0. There is evidence to conclude that the mean arsenic concentration in the drinking water in rural Arizona is differ from the mean arsenic concentration in metropolitan Phoenix. Furthermore, the mean arsenic is higher in rural of Arizona. P value is approximate P=0.016

Case II

Sample size can be approximated by OC curves Only for the case that 1= 2

Where and

Ex.

2|| 0

d 12* nn

Case III: Paired t- test

A special case of the two-sample t-test. This test is used when the observations on the two populations of interest are collected in pairs.

Each pair of observations is taken under homogeneous condition.

Ex. We are interested in comparing two different types of tips for a hardness-testing machine.Tip1 Tip2

Sheet Metal

Comparing the depth of the depression caused by the tips

Tip1 Tip2Pair t-test

2 sample test

Case III: Paired t-test

Paired t-test Hypothesis

Test Statistic


0211

0210

:

:

H

H

01

00

:

:

D

D

H

H

nS

DT

D

00

1,2/01,2/0 nn ttortt


Ex. An article in the journal of Strain Analysis compares several methods for predicting the shear strength for steel plate girders. Data for two of these methods, Karlsruhe and Lehigh procedures, when applied to nine specific girders are shown in table. We wish to determine whether there is any difference between the two methods.


1. Parameter of interest: the difference in mean shear strength between the two methods µD=µ1-µ2

2. H0: µD=0, H1: µD≠0

3. =0.05

4. Test statistic:

5. We should reject H0 if

6. Calculate t0

7. T0 =6.08>2.306, we conclude that the strength prediction methods yield different results. Specifically, the data indicate that the Karlsruhe method procedures, on the average, higher strength predictions than does the Lehigh method. P value for t0 = 6.08 is P=0.0003

nS

DT

D

00

306.2306.2 8,025.008,025.00 ttortt

08.691351.0

02739.00

t


Confidence Interval


Ex

Inference on the variance two normal distribution Hypothesis

Test Statistic


211

210

:

:

H

H

22

21

0 S

SF

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f,,

1,,1

Inference on the variance two normal distribution Ex

Inference on the variance two normal distribution

Inference on the variance two normal distribution Sample size: can be approximated by OC

curve Only for the case that n1=n2=n Where Ex

2

1

Inference on the variance two normal distribution Confidence Interval on the ratio of two

variances

Inference on the variance two normal distribution Ex.

Inference on the variance two normal distribution

Test on two population proportion Hypothesis

Test Statistic


211

210

:

:

ppH

ppH

21

21

21

210

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)11

)(ˆ1(ˆ

ˆˆ

nn

XXP

nnPP

PPZ

2/02/0 zzorzz

Test on two population proportionEx.

Test on two population proportion

Test on two population proportion Type II error

Test on two population proportion Sample size

For one sided, replace /2 by

Test on two population proportion Confidence Interval on the difference in

Population proportions

Test on two population proportion Ex

Test on two population proportion

Homework

Using Minitab program to find the conclusion of these problems.

1. An article in solid state technology describes an experiment to determine the effect of the C2F6 flow rate on the uniformly of the etch on a silicon wafer used in integrated circuit manufacturing. Data for two flow rate are as follows:

a) Does the C2F6 flow rate affect average etch uniformity? Use =0.05

b) What is the P-value for the test in a)

c) Does the C2F6 flow rate affect the variability in etch uniformity? Use =0.05

d) Draws box plots to assist in the interpretation of the data from this etch uniformity.

C2F6 flow rate

observation

1 2 3 4 5 6

125 2.7 4.6 2.6 3.0 3.2 3.8

200 4.6 3.4 2.9 3.5 4.1 5.1

Homework

2. A computer scientist is investigating the usefulness of two different design languages in improving programming task. Twelve expert programmers, familiar with both languages, are asked to code a standard function in both language, and the time (in minutes) is recorded. The data follow:

a) Is the assumption that the difference in coding time is normally distributed reasonable?

b) Find P-value for the test in a)

c) Find a 95% confidence interval on the difference in mean coding times. Is there any indication that one design language is preferable?

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Test of Hypotheses: Two Sample