Lecture 7 QM Introduction to Hypothesis Testing SIX

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Hypothesis Testing in Statistics:

An Introduction

Lecture 7

Lecturer : Dr. Dwayne Devonish

MGMT 2012: Introduction to Quantitative MethodsLearning Objectives

• Students should be able to:

� List the steps of hypothesis testing,

�Distinguish between null and alternative

hypotheses,

�Distinguish and choose between different types

of one-sample hypothesis tests (z-test versus t-

test)

�Work out the critical values of these tests to

distinguish between rejection and nonrejection

decisions

Hypothesis Testing• Oftentimes, quantitative analysts want to make decisions or

conclusions about populations on the basis of the sample

data collected.

• Hypothesis testing allows these analysts to make claims

(statements) about certain characteristics of the population

(known as parameters such as population means or

proportions), and apply statistical tests on sample data to

assess how plausible (or likely to be true) these statements

are.

• These statistical tests provide sample evidence that leads

one to either accept or reject a particular hypothesis.

• Essentially, the researcher makes generalisations about the

population (parameters) on the basis of sample data

(statistics).

Steps in Hypothesis Testing• Step 1: State two opposing hypotheses (null vs

alternative hypotheses)

• Step 2: Determine the appropriate statistical test (and

associated distribution) to assess the hypotheses

• Step 3: Determine the ‘critical values’ that identify

nonrejection (acceptance) and rejection regions for the

null hypothesis

• Step 4: Apply the chosen statistical test to

obtain/compute a test-statistic (a.k.a. sample evidence)

• Step 5: Check to see if sample evidence or test statistic

falls within the acceptance or rejection regions

• Step 6: Make conclusion about the hypothesis to be

accepted or rejected.

Step 1: Null vs Alternative Hypothesis• Null Hypothesis: Ho : This hypothesis is always stated in

statistical terms, using population parameters .

• Ho: µ = 500 (where µ = 500 is a claim regarding the

population mean - e.g. mean income of all public

sector workers) is $500 per week.

• The null hypothesis is assumed to be true unless the

statistical test provides evidence that contradicts it.

• The null hypothesis normally represent the status quo,

and always specifies that a population parameter

equals a specific value.

• The null hypothesis is always the hypothesis that is

under investigation (hence, one seeks to determine if it

can be rejected or not).

Examples of Null Hypotheses

A. “Years ago, the average monthly salary of a senior

public sector worker was BDS$1,200”.

• Ho: µ = 1200

B. “Research has reported that approximately 60% of

men who smoke are likely to develop health

problems”.

• Ho: p = .60

(where ‘p’ is the population parameter for population

proportion of male smokers who develop health

problems

The null hypothesis always contains an ‘equal sign’ (=)

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Alternative Hypothesis• The alternative hypothesis Ha : This is always the opposite

of the Ho. If Ho is rejected, then there is evidence to

support Ha. This hypothesis is the researcher’s actual or

primary hypothesis that he or she wants to ‘prove to be

true’ based on sample (statistical) evidence.

• It never contains an equal sign but specifies a population

parameter: is not equal to ( ≠ ), greater than (>), or less

than ( < ) a specific value.

• E.g. HA : µ ≠ 500 (population mean is not equal to 500)

• HA : µ > 500 (population mean is greater than 500)

• HA : µ < 500 (population mean is less than 500)

• The type (or sign) of alternative hypothesis chosen

depends on what the researchers wants to prove within a

given scenario.

Example 1

• It has often been claimed in the past that the averagewaiting time to conduct business at Rogel Bank ofBarbados is 10 minutes. If we were to collectinformation today, can we say that this claim is stilltrue? (or can we say this claim is no longer true).

• Ho: µ = 10

• Ha: µ ≠ 10 (this is known as a two-sided hypothesis)

• Two-sided/tailed alternative hypothesis indicates twoopposite possibilities to the null hypothesis: i.e. themean waiting time today could be either higher orlower than 10 minutes.

Example 2• It has often been claimed in the past that the average

waiting time to conduct business at Rogel Bank of

Barbados is 10 minutes. If we were to collect

information today, can we say the mean waiting time,

due to improved technology, is less than 10 minutes.

• Ho: µ = 10

• Ha: µ < 10 (this is known as one-sided hypothesis)

• One-sided/tailed alternative hypothesis indicates only

one possibility to the null hypothesis; in this case, that

the mean waiting time is less than 10 minutes.

• Note: Ha: µ > 10 is also a one-sided hypothesis which

suggests that mean is greater than 10 minutes (one

possibility).

Step 2: Statistical Test and Distribution

• In order to test the hypotheses in the prior scenario,

the analyst must determine the appropriate statistical

test to use.

• There are many statistical tests that we will cover in

hypothesis testing but we will use examples from the

most basic types of statistical tests which are used to

test basic hypothesis testing scenarios:

• ONE-SAMPLE TESTS

One-Sample Tests

• One-sample tests can be used to examinewhether a population parameter such as apopulation mean (µ) or populationproportion/percentage (p) is a specific value (ornot).

• These tests are called one-sample tests becausethey examine whether claims about a populationmean or proportion is plausible or not on thebasis of statistical data or evidence derived froma single sample of observations collected.

Types of One-Sample Tests

A. One sample tests can be used for testing a populationmean: one-sample tests for a population mean, µ

B. One sample tests can be used for testing a populationproportion: one-sample tests for a populationproportion, p.

• Very soon, we will briefly look at one-sample tests forthe population mean to explain how the first threesteps of hypothesis testing (in an earlier slide) areimplemented.

• However, the next lecture will deal more deeply withtheir calculation and full use in various hypothesistesting scenarios (based on the last three steps).

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Remember Steps in Hypothesis Testing• Step 1: State two opposing hypotheses (null vs






null hypothesis

• Step 4: Apply the chosen statistical test to

obtain/compute a test-statistic (a.k.a. sample evidence)

• Step 5: Check to see if sample evidence or test statistic

falls within the acceptance or rejection regions



One-Sample tests for the Population

Mean• There are two (2) popular one-sample tests for the

population mean.

• The Z-test (based on z-distribution):

� This test is used under the following conditions:

1. If the sample size is large (n ≥ 30)

OR

2. If the population standard deviation is known (σ)

Either 1 or 2 alone or if both hold, z-test is applied.

NB: If population standard deviation is known but sample

size is small (n < 30), z-test is still used but the sample

standard deviation is used as an estimate/substitute.

One Sample tests for the Population

Mean

• The T-test (based on Student t-distribution):

�This test is used under the following conditions:

1. The sample size is small (n < 30)

AND

2. The population standard deviation is unknown

(σ = ??)

Both 1 and 2 must hold for t-test to be applied.

One Sample Z-test and T-test• Both tests assume that the population that one is generalising

to is a ‘normal’ population.

• The choice between these two tests rests on information

available in specific hypothesis-testing scenarios.

• Ask yourself two questions:

a) Is the sample size large (30 or greater)?

b) Is the population standard deviation known or unknown.

• Then, choose the appropriate test.

• These tests are used to determine whether a population mean

(µ) is a specific value (or not) on the basis of a sample mean

(x)̄ derived from a set of sample observations (data).

• NB: The z-test alone is used to test for population

proportions.

Remember Steps in Hypothesis Testing

�LET’S SEE HOW THESE TESTS WORK IN THE FIRST THREE

STEPS OF HYPOTHESIS TESTING

• Step 1: State two opposing hypotheses (null vs






null hypothesis

Example A (step 1 and 2)• It has been said that the average monthly salary of all public

sector workers is BDS $5,000. A random sample of 40 public

sector workers was examined and a mean salary of BDS

$4800 and standard deviation of BDS $1500 were found.

Test whether the population mean has changed.

• Step 1: Hypotheses:

• Ho: µ = 5000

• Ha: µ ≠ 5000 (2-tailed)

• Step 2: Test??

a) Is the sample size large (30 or greater): YES

b) Is the population standard deviation known or unknown:

UNKNOWN

ANSWER: Z-TEST

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Example B (step 1 and 2)• It has been said that the average monthly salary of all public

sector workers is BDS $5,000 with a standard deviation of BDS

$2200 was found. A random sample of 20 public sector

workers was examined and a mean salary of BDS $4800 was

found. Test whether the population mean salary has

decreased.


• Ho: µ = 5000

• Ha: µ < 5000 (1-tailed)

• Step 2: Test??

a) Is the sample size large (30 or greater): NO


KNOWN (= $2200)

ANSWER: Z-TEST

Example C (step 1 and 2)• It has been said that the average monthly salary of all public

sector. workers is BDS $5,000. A random sample of 20 public

sector workers was examined and a mean salary of BDS

$6800 and a standard deviation of $1200 were found. Test

whether the population mean has increased.


• Ho: µ = 5000

• Ha: µ > 5000 (1-tailed)

• Step 2: Test??

a) Is the sample size large (30 or greater): NO


UNKNOWN

ANSWER: T-TEST

Step 3: Determine Critical Values for

Rejection/Acceptance Regions • Critical values are associated with the distribution of the

statistical tests selected in step 2 (z-tests vs t-tests) and separate

acceptance and rejection regions for the Ho.

• They are based on which statistical tests you are using and the

level of significance or alpha level (α).

• For the Z-test: The critical values are determined on the basis

of the alpha level or level of significance.

• The alpha level represents the probability of rejecting the null

hypothesis when it is in fact true (known as Type 1 error) –

although we are testing hypotheses, we can still make errors

because we are using sample (i.e. incomplete) data.

• So we always have to set an alpha level based on the level of

error we are willing to tolerate (usually 5% or 0.05 but can be

1% or .01; or 10% or .10).

Z-test: Critical Values• Critical values for the z-test distribution are also linked

to whether a hypothesis testing scenario is one-tailed

or two-tailed.

• Critical values are set on the distribution to separate

two key regions based on the alpha level: a rejection

region or nonrejection (acceptance) region for Ho.

• The alpha region can be known as the rejection region

where Ho is rejected (recall that alpha is probability of

rejecting Ho when it is true). The other part of

distribution region is the nonrejection region.

• The alpha region is usually on the tail(s)of the

distribution; split between both tails in a two-sided

scenario; or in one tail alone in a one-tailed scenario.

UPPER TAILUPPER TAIL

00 UPPER CVUPPER CV

Reject H0Reject H0Do Not Reject H0Do Not Reject H0

zz

Reject H0Reject H0

LOWER CVLOWER CV

�� Critical values (CV) can be lower (negative) values or upper Critical values (CV) can be lower (negative) values or upper (positive) values depending on 1(positive) values depending on 1--tailed or 2tailed or 2--tailed hypotheses.tailed hypotheses.

Samplingdistribution

of z-test


of z-test

ZZ--DISTRIBUTION CURVEDISTRIBUTION CURVE

LOWERΤΑΙLLOWERΤΑΙL

�� This is an example of a twoThis is an example of a two--tailed scenario, you see tailed scenario, you see lower and upper CVs within which nonrejection of Ho lies.lower and upper CVs within which nonrejection of Ho lies.

Prior Z-test Example: One-tailed• It has been said that the average monthly salary of all public


sector workers was collected and a mean salary of BDS $4800

and standard deviation of BDS $1500 were found. Test at a 5%

level whether the population mean has decreased.

• Ho: µ = 5000

• Ha: µ < 5000 (1-tailed)

• The alpha level is 5% or 0.05. Given the 1-tailed nature of

scenario, the 5% of the left tail of z-distribution is shaded as the

rejection region (where Ho should be rejected): α = .05 = critical

value of -1.645 (see distribution table).

• α is on the left tail because we are testing whether the

population mean is less (on the left of zero on distribution). If

we were testing ‘greater than’; 5% of the right tail is shaded as

rejection region (critical value = 1.645).

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Z-DISTRIBUTION (RIGHT OR LEFT TAIL)

Critical values will be

negative if lower limit

of left tail was used

( -1.645, -1.960, etc)

‘GUIDESHEET FOR Z-TEST’

Critical values of the z-distribution

One-Tailed

Tests

α Left Tailed

Test

Right

Tailed Test

Two Tailed

Test

10% -1.28 +1.28 +/- 1.645

5% -1.645 +1.645 +/- 1.96

2% -2.05 +2.05 +/- 2.33

1% -2.33 +2.33 +/- 2.575

α = .05α = .05

00−zα = −1.645−zα = −1.645

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

zz


of z-test


of z-test

Lower OneLower One--Tailed Z TestTailed Z Test

�� Critical value of Critical value of --1.645 is always for .05 one1.645 is always for .05 one--tailed.tailed.�� Critical value of Critical value of --1.645 is always for .05 one1.645 is always for .05 one--tailed.tailed.

�� Lower (left) oneLower (left) one--tail tests always have negative critical tail tests always have negative critical values, higher (right) onevalues, higher (right) one--tail tests have positive ones.tail tests have positive ones.


Prior Z-test Example: Two-tailed• It has been said that the average monthly salary of all public

sector workers is BDS $5,000. A random sample of 40 public

sector workers was collected and a mean salary of BDS

$4800 and standard deviation of BDS $1500 were found.

Test at a 5% level whether the population mean has

changed.

• Ho: µ = 5000

• Ha: µ ≠ 5000 (2-tailed)

• The alpha level is 5% or 0.05. Given 2-tailed nature of

scenario, the 5% must be shared between left and right tails

of z-distribution. Hence, α/2 = 2.5% or .025 in each tail.

Hence, there are lower and upper critical values for .025 in

each tail which are -1.96 to 1.96 (see distribution table).

Within this range lies the nonrejection region of Ho.

α/2 = .025α/2 = .025

00 1.961.96


zz

Reject H0Reject H0

-1.96-1.96

�� TwoTwo--tailed tests provide a range of lower and upper tailed tests provide a range of lower and upper critical values (critical values (--11..96 96 to to 11..9696).).


of z-test


of z-test

TwoTwo--Tailed Tailed ZZ--TestsTests

α/2 = .025α/2 = .025

�� The alpha for twoThe alpha for two--tailed tests are always divided by tailed tests are always divided by 2 2 to be shared between the two tails equally.to be shared between the two tails equally.

T-Test: Critical Values

• The t-test is based on Student’s t-distribution and

the determination of its critical values are different.

• Critical values are based on two factors:

� Alpha level (again if 2-tailed, divide α by 2; if 1-

tailed, determine whether right or left tail contains

the alpha)

� Degrees of freedom (Sample size minus 1)

• Once you determine these factors, the critical values

can be found within the t-distribution tables.

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Prior T-test example (2-tailed)• It has been said that the average monthly salary of all public


sector workers was collected and a mean salary of BDS


at the 5% level whether the population mean has changed.

• Ho: µ = 5000

• Ha: µ ≠ 5000 (2-tailed)

• The alpha level is .05, two-tailed. So each tail has 2.5% or

.025. T-test distribution tables are easy to read . Look for

two-tailed section under 5% (where α/2), and then go down

to the relevant degrees freedom (remember sample size

minus one or 20-1 = 19). The critical values must be written

as lower and upper values: -2.093 to 2.093. Within this

range lies the nonrejection region of Ho.

T-TEST TABLES

α/2 = .025α/2 = .025

00 2.0932.093


tt

Reject H0Reject H0

-2.093-2.093

�� TwoTwo--tailed tests provide a range of lower and upper tailed tests provide a range of lower and upper critical values (critical values (--2.093 to 2.093).2.093 to 2.093).


of t-test


of t-test

TwoTwo--Tailed Tailed TT--TestTest

α/2 = .025α/2 = .025

�� The alpha for twoThe alpha for two--tailed tests are always divided by tailed tests are always divided by 2 to be shared between the two tails equally.2 to be shared between the two tails equally.

2nd T-test example (1-tailed)• It has been said that the average monthly salary of all public


sector workers were collected and a mean salary of BDS


at the 5% level whether the population mean has reduced.

• Ho: µ = 5000

• Ha: µ < 5000 (1-tailed)

• The alpha level is .05 for 1-tail. It is .05 only on the left tail.

Look for one-tailed section under 5%, and then go down to

the relevant degrees freedom (remember sample size minus

one or 20-1 = 19). The critical value is a lower or negative

value, -1.729, given it is on left-side of distribution.

T-TEST TABLES

α = .05α = .05

00−tα = −1.729−tα = −1.729

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

tt


of t-test


of t-test

Lower OneLower One--Tailed TTailed T--TestTest

�� Critical value of Critical value of --1.729 is always for .05 one1.729 is always for .05 one--tailed.tailed.�� Critical value of Critical value of --1.729 is always for .05 one1.729 is always for .05 one--tailed.tailed.



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© 2002 Prentice-Hall, Inc. Chap 9-37

Summary of Alpha level

and the Critical Values for Z- and T-

Tests

H0: µµµµ = = = = 30 (or µµµµ ≥ ≥ ≥ ≥ 30)

Ha: µµµµ < 300

0

0

H0: µµµµ = = = = 30 (or µµµµ ≤≤≤≤ 30)

Ha: µµµµ > 30

H0: µµµµ = = = = 30

Ha: µµµµ ≠≠≠≠ 30

αααα

αααα

αααα/2

Critical

Value(s)

Rejection Regions

Next Lecture• This was only an introduction to hypothesis testing; there

are more steps until we are complete:

• Step 4: Apply/compute the chosen statistical test to

obtain a test-statistic (or aka sample evidence)

• Step 5: Check to see if sample evidence falls within the

acceptance or rejection regions



• These final three steps allow us to actually compute the

sample/statistical evidence to see which hypotheses are

correct. We will go further into one-sample and two

sample tests next session.

Documents

Lecture 7 QM Introduction to Hypothesis Testing SIX