Two Means Test

8/12/2019 Two Means Test

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Confidence Interval estimate for

1 - 2


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Parameters/Estimates (IndependentSamples)

Parameter: Estimator: Estimated standard error:

N1 = 100, Sample Mean = 50, s 1 = 6N2 = 150, sample mean = 65, s 2 = 8Determine the 90% confidence interval for the

difference between population mean.

21 Y Y

2

2

2

1

2

1

n

S

n

S


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Problem 1

Two cities are separated by a river. The chamber ofcommerce in city A claims that the average family incomeof that city is Rs.500/- more than that in city B. Thechamber of commerce in city B disputes this, and astatistician is hired to settle the argument. He attacks theproblem by estimating the true difference between thevarious family income in the two cities. One randomsample is drawn from each city, and the following resultsare obtained :From City A 100 families are selected and the average

family income found to be Rs. 1500 with variance 2260.From city B 120 families are selected with average incomeRs. 1400 and variance 2100. Obtain a 99% confidenceinterval estimate for the difference in mean.


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Small-Sample Test for Normal Populations

Case 1: Common Variances ( s 12 = s 22 = s 2) Null Hypothesis : Alternative Hypotheses :

1-Sided: 2-Sided :

Test Statistic: (where S p2 is a pooled estimate of s 2)

0210 : H

021: A H

021: A H

2)1()1(

11

)(

21

222

2112

21

2

021

nnS nS n

S

nnS

y yt p

p

obs


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Small-Sample Inference for Normal Populations

Case 2: s 12 s 22

Dont pool variances:

Use adjusted degrees of freedom (SatterthwaitesApproximation) :

2

2

2

1

2

1

21 nS

nS S y y

11

*

2

2

2

22

1

2

1

21

2

2

22

1

21

n

nS

n

nS

nS

nS


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Problem 2

The makers of ink cartridges for color ink-jetprinters have developed a new system for storingthe ink. They think the new system will result in alonger lasting product. In order to determinewhether this is the case, a test was developed inwhich a sample of 35 of the new cartridges wasselected. They were put in a printer, and the testpages were run until the cartridge was empty.

The same thing was done for a sample of 32 original cartridges. The following data wereobserved.


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New cartridge mean 288 pages, standard deviation = 16.3 pages

Existing cartridge Mean 279 pages, standard deviation = 15.91 pages

Based on the sample data and a significance levelequal to 0.1, determine if the new system willresult in a longer lasting product.


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Problem

A reading test is given to an elementary schoolclass that consists of 12 south Indian childrenand 10 north Indian children. The results ofthe test are

North Indian : Mean 74, standard deviation 8South Indian : Mean 70, standard deviation 10Is the difference between the means of the two

groups significant at .05 level?


9/15

Behavioral research has developed an index designed tomeasure managerial success. The index measure (in 100point scale) is based on the managers level of time in theorganization and his/her level within the firm; the higherthe index, the more successful the manager. Suppose aresearcher wants to compare the average success index forthe two groups of managers at a large manufacturing plant.Manager in group 1 engage a high volume of interactionswith people outside the managers work unit. Managers ingroup 2 rarely interact outside their work unit. Independentrandom samples of 12 and 15 managers are selected fromthe groups 1 and 2, respectively, and the success index ofeach recorded.


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The results are as followsGroup 1: 65 58 78 60 68 69 66 70 53 71

63 63.

Group 2: 62 53 36 34 56 50 42 57 46 6848 42 52 53 43Use the data in the table to estimate the mean

difference between the success indexes ofmanagers in two groups. Use a 95% confidenceinterval.

Is the difference between the means of the twogroups significant at .05 level?


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Test Concerning D

Null Hypothesis : H 0: D=0

Test Statistic:

n s

d t

d

obs


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Problem

HomeGrocer.com is an online grocery store in theSeattle area that has more than 10000customers. The following table reports May 1999

prices for a shopping list of eight items fromHomeGrocer.com and local supermarket.At .05 level of significance, is there any evidence of

a difference in the average price for products

purchased from HomeGrocer.com and thesupermarket.

Find the p-value.


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Products Home-Grocer Super-Markets

Tide Detergent 6.99 6.99

Cookies 3.29 3.49

Formula cleaner 2.59 2.69

Pampers Diapers 10.79 10.99Coke Classic 3.99 3.59

Colgate Toothpaste 3.49 3.49

Tropicana Orange Juice 3.59 3.49

Cereal 4.29 3.99


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Problem

Marketing strategies would like to predict consumerresponse to new products and their accompanyingpromotional schemes. Consequently, studies theexamine the differences between buyers and

nonbuyers of a product are of interest. One studyshowed that both the mean threshold size (number ofpersons) and the mean threshold income weresignificantly larger for purchaser than nonpurchaser. Asimilar study of utilized independent random sample of

size 20 and yielded the data on the age of thehouseholder primarily responsible for buyingtoothpaste.


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Purchaser : 23 34 35 44 46 52 28 48 2834 33 52 41 32 34 49 50 45 29 59

Nonpurchaser : 28 22 33 44 55 31 45 5460 63 52 53 58 52 66 35 25 48 5961

Do data present sufficient evidence toconclude that there is a difference in themean age of purchasers and nonpurchasers?Use = .10

Find the observed significance level of thetest.

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Two Means Test