Quantitative Business Methods for Decision Making

Estimation and Estimation and Testing of Testing of

HypothesesHypotheses

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Lecture OutlinesLecture Outlines

Estimation Confidence interval for estimating

means Confidence interval for predicting a

new observation Confidence interval for estimating

proportions

403.6 3

Lecture Outlines (con’t)Lecture Outlines (con’t)

Hypothesis Testing Null and alternative hypotheses Decision rules (Tests) and their level of

significance Type I and Type II errors Tests of hypotheses for comparing

means Tests of hypotheses for comparing

proportions

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Estimating a Population MeanEstimating a Population Mean

Population mean is estimated by , the sample mean

Standard error of , i.e.

will decrease as n gets large.

x

n

ss

x

x

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Confidence Interval for Confidence Interval for Estimating if is Estimating if is

knownknown

With a 95% degree of confidence is estimated within ( )

written as Or more accurately

by

nx

x2

X

x2X

x2

X

x96.1 X

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Confidence Interval for if Confidence Interval for if is not known is not known

Use instead of ,

remember , and

“t” is 95th% percentile of the t distribution with (n-1) degrees of

freedom.

x96.1 x

xtsx

nx

n

s

xs

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An IllustrationAn Illustration

Suppose n= 26. Then degrees of freedom (d.f.) = n-1 = 25.

A two-sided degree of C.I. is computed

by But, for a one-sided 95% C.I. , t = 1.711

instead of 2.064

%95

x2.064sx

403.6 8

Assumptions and Sample Size Assumptions and Sample Size forfor

Estimation of the meanEstimation of the mean

The population should be normally (at least close to) distributed. If skew, then median is an appropriate measure of the center than themean.To estimate mean with a specified margin oferror (m.e.), take a random sample of size n

large size. 2

22

..em

zn

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Prediction Interval for a Prediction Interval for a New Observation on XNew Observation on X

n

11ts x

Prediction Interval for a new observation is given by

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Confidence Interval for a Confidence Interval for a Population ProportionPopulation Proportion

Let denote the proportion of items in a

population having a certain propertyAn estimate of is the binomial proportion: , What is ?

For a C.I. for , use

ptsp

n

Xp

X

403.6 11

Confidence Intervals for Confidence Intervals for the Proportion (con’t)the Proportion (con’t)

For estimating ,“t” is the percentile of the

t-distribution with (equivalently, percentile of the standard normal distribution), and s.e. of p is

df

n

pp )1(sp

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Hypotheses Testing Hypotheses Testing

The hypothesis testing is a methodology for proving or disproving researcher’s prior

beliefs. Statements that express prior beliefs are framed as alternative hypotheses. Complementary statement to an alternative hypothesis is called null hypothesis.

403.6 13

Null and AlternativeNull and Alternative

Ha: Researcher’s belief that are to be tested (alternate hypothesis)

H0: Complement of Ha (Null hypothesis)

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Statistical Decision

A decision will be either: Reject H0 (Ha is proved)

orDo not reject H0 (Ha is not proved)

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Hypothesis Testing Methodology Hypothesis Testing Methodology for the meanfor the mean

Depending upon what an investigator believes a priori, an alternative hypothesis is formulated to be one of the followings:1.

2.

3.

0a :H

0a :H

0a :H

one-sided

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A Test StatisticA Test Statistic

Regardless of what an alternative hypothesis

about the mean is formulated, the decision

rule is defined by a t- statistic:

x

0

S

Xt

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Decision Rules for Testing Decision Rules for Testing Hypotheses About the MeanHypotheses About the Mean

Hypotheses Decision Rule

1. Ho: = o At level of significance, reject Ho in favor of Ha if

Ha: o either t-statistic 2t or -

2t

2. Ho: o At level of significance, reject Ho in favor of Ha if

Ha: o t-statistic t

3. Ho: o At level of significance, reject Ho in favor of Ha if

Ha: o t-statistic -t

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Type I and Type II ErrorsType I and Type II ErrorsDecision taken is

Accept H0 Accept H1

----------------------------------------------------------------------- Suppose correct Type I Error H0 is true decision (wrong decision)------------------------------------------------------------------------ Suppose Type II Error correct H1 is true (wrong decision) decision

Type I error : reject H0 if H0 is true.

Type II error : Do not reject H0 when H0 is false.

403.6 19

Comparing Two MeansComparing Two Means

The reference number is a specified amount for comparing the

difference between two means. There are two distinct practical

situations resulting in samples on X and Y.

X population Y population X ,

X Y , Y

Consider

1. H0: X - Y = 2. H0: X - Y 3. H0: X - Y

Ha: X - Y , Ha: X - Y , Ha: X - Y .

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Two Sampling Designs Two Sampling Designs

•Paired Sample•Two independent Samples

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Paired SamplePaired Sample Two variables X and Y are observed for

each unit in the sample to measure the same aspect but under two different conditions.

Thus, for n randomly selected units, a sample of n pairs (X, Y) is observed.

Compute differences: X1-Y1= d1, X2-Y2= d2, etc. and then mean

Compute Sd of differences

dd

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Paired Samples (con’t)Paired Samples (con’t)

Compute

t-statistic:

dS

d statistic-t

n

sdd

S

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Paired Samples (con’t)Paired Samples (con’t)

•Reject H0 if absolute value of t-statistic is more than the desired percentile of the t-distribution.

•Alternatively, find the p value of the t-statistics and reject H0 if the p value is less than the desired significance level.

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Two Independent Two Independent (Unpaired) Samples(Unpaired) Samples

Populations of variables X and Y (for example, males salary X and females salary Y). Take samples independently on X and

Y. Compute Compute pooled standard deviation

yx S,Y,S,X

2

S1S1nS

21

2y2

2x1

nn

n

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Unpaired Samples Unpaired Samples (con’t)(con’t)

Compute

Finally, compute t-statistic=

Use p value to reach a decision

21

x n

1

n

1SSX of SE

yY

y

xS

YX

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Comparing ProportionsComparing Proportions

To estimate in a 95% C.I., compute,

21

2

22

1

1121

1196.1

n

pp

n

pppp

21 and

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Comparing Two Comparing Two ProportionsProportions

For testing hypothesis about the difference , compute

and

t-statistic=

21

n

pnpnpnnn 2211

21 ,

n

pp

pp

)1(21