chapter 10 stat

7/29/2019 Chapter 10 stat

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BUS 173

Chapter 10: Inference about the difference between two population means: 1 and

2 known and unknown

1and 2 known

1 and 1 are population characteristics of a population1.2and 2 are population characteristics of a population2.Difference between two population means = 1 - 2x1-bar = sample mean from a sample on population1x2 - bar = sample mean from a sample on population2

The point estimator of the difference between the two population means is the differencebetween the two sample means. x1-barx2bar

Standard Error of x1-barx2-bar, x1-barx2-bar= (12/n1+ 2

2/n2)

If both populations have a normal distribution and if the sample sizes are large enough,

then the central limit theory allows us to approximate the difference between the using

a normal distribution.

The Margin of Error = Z/2x1-barx2-bar= Z/2 (1

2/n1+ 2

2/n2)

Where 1- is the confidence coefficient

Given this margin of error, internal estimate of the difference between the two

population means, x1-barx

2bar Z

/2(

1

2/n

1+

2

2/n

2)

Where 1-is the confidence coefficientHypothesis test about 1-2:

There can be three forms of hypothesis tests:

H0 : 1-2 D0Ha : 1-2 < D0

H0 : 1-2 D0Ha : 1-2 < D0

H0 : 1-2 = D0

Ha : 1-2 D0


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In many application of hypothesis tests involving difference D0 = 0. For example, in a

two tailed test involving D0 = 0, the null hypothesis is 1=2, rejection of this would lead

to 1 and 2 are not equal.

Test statistic for 1-2 hypothesis test, 1 and 2 known

Z = ((x1-barx2bar)D0)/ (12/n1+ 2

2/n2)

Interference about the difference between two population means: 1and 2

unknown

When the values of the population standard deviation are not known, we use the t-

distribution score instead of the standard normal distribution Z score.

Internal estimate for 1 and 2 unknown case is, x1-barx2bar t/2 (12/n1+ 2

2/n2)

Where 1-is the confidence coefficientThe problem that we face here is with finding the appropriate degrees of freedom incalculating the t static,

df = (s12/n1 + s2

2/n2)

2/ ((1/(n1-1)(s1

2/n1)

2+ (1/(n2-1)(s2

2/n2)

2)

Test static for 1 and 2 unknown case,

t = ((x1-barx2bar)D0) / (s12/n1+ s2

2/n2)

Where degrees of freedom is to be found using the above mentioned formula

Interference about the difference between two population means: Matched Samples

When we use the same sample assess the difference between two things, we need to use

the matched sample method. On a matched sample we use the difference data for the teststatistic, d-bar = di / n

Sd = ( (d1- d-bar)2

/ (n-1))

Test static for matched samples, t = (d-bar - d) / (Sd/n)

Where the degrees of freedom = n-1

Interference about the difference between two population proportions

When considering the differences between two population proportions p1 and p2, we

consider the following test statistics:

p1p2 point estimator = p1-barp2-bar


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Standard Error of p1-barp2-bar

p1-barp2-bar= ((p1(1-p1)/n1) + (p2(1-p2)/n2)

for Margin of Error we cannot use Z/2 or p1-barp2-barbecause p1 and p2 are unknownusing the sample proportion p1-bar to estimate p1 and sample proportion p2-bar to

estimate p2, the margin of error is as follows:

M.E. = Z/2 ((p1-bar(1-p1-bar)/n1) + (p2-bar(1-p2-bar)/n2)

Internal Estimate would be

p1-barp2-bar Z/2 ((p1-bar(1-p1-bar)/n1) + (p2-bar(1-p2-bar)/n2)

Where 1-is the confidence coefficientStandard Error of p1-barp2-bar when p1= p2 = p

p1-barp2-bar= ((p(1-p)/n1) + (p(1-p)/n2)

= (p(1-p)(1/n1+1/n2))

With p unknown, pooled estimator of p when p1= p2 = p

p-bar = (n1p1-bar + n2 p2-bar) / (n1+n2)

Pooled estimator of p is the weighted average of p1-barand p2-bar

Test statistics for tests about p1p2

Z = (p1-barp2-bar) / (p-bar(1 - p-bar) (1/n1+1/n2))

chapter 10 stat

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