copyright © 2003 mcgraw-hill ryerson limited. all rights reserved. 10 - 1

10 - 1

Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved.

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Define null and alternative hypothesis and hypothesis testing

Define Type I and Type II errors

Describe the five-step hypothesis testing procedure

Distinguish between a one-tailed and a two-tailed test of hypothesis

When you have completed this chapter, you will be able to:

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Conduct a test of hypothesis about a population mean

Conduct a test of hypothesis about a population proportion

Explain the relationship between hypothesis testing and confidence interval estimation

Compute the probability of a Type II error, and power of a test

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Terminology

Hypothesis…is a statement about a population distribution such that:

ExamplesExamples …the mean monthly income for all systems analysts is $3569.

…the mean monthly income for all systems analysts is $3569.

…35% of all customers buying coffee at Tim Horton’s return within a week.

…35% of all customers buying coffee at Tim Horton’s return within a week.

(i) it is either true or false, but never both, and(ii) with full knowledge of the population data, it is possible to identify, with certainty,

whether it is true or false.

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Terminology

…is the complement of the alternative hypothesis.

We accept the null hypothesis as the default hypothesis. It is not rejected unless there is

convincing sample evidence against it.

Null Hypothesis Ho

Alternative Hypothesis H1…is the statement that we are interested in proving

.

It is usually a research hypothesis.

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State the decision ruleState the decision rule

Identify the test statisticIdentify the test statistic

Do NOT reject H0Do NOT reject H0 Reject H0 and accept H1

Reject H0 and accept H1

Compute the value of the test statistic and make a decision

Compute the value of the test statistic and make a decision

Step 1Step 1

Select the level of significanceSelect the level of significanceStep 2Step 2

Step 3Step 3

Step 4Step 4

Step 5Step 5

Hypothesis Testing Hypothesis Testing

State the null and alternate hypothesesState the null and alternate hypotheses

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When a decision is based on analysis of sample data

and not the entire

population data, it is not possible to make a correct decision all the time.

Our objective is to try to keep the probability of making a wrong decision

as small as possible!

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Let’s look at the Canadian legal system for an analogy...Let’s look at the Canadian legal system for an analogy...

1. …the accused person is innocent

2. …the accused person is guilty

Two hypotheses:

After hearing from both the prosecution and the defence, a decision is made, declaring the accused either:

After hearing from both the prosecution and the defence, a decision is made, declaring the accused either:

Innocent! But do the courts always make the “right”

decision?Guilty!

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Person is “innocent”

Person is “guilty”

Person is declared

’not guilty’

Person is declared “guilty”

Correct DecisionCorrect Decision

Correct DecisionCorrect Decision

Error

Error

H0: person is innocent H1: person is guilty

H0 is true

H1 is true Type II Error

Type I Error

Court Decision

Reality

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Terminology

Level of Significance…is the probability of rejecting the null hypothesis

when it is actually true, i.e. Type I Error

…accepting the null hypothesis when it is actually false.

Type II Error

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Terminology

Test Statistic…is a value, determined from sample information,

used to determine whether or not to reject the null hypothesis.

Critical Value…is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

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TestsTests

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0Critical z

=

rejection region

1- =

acceptance region

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0

=

rejection region

1- =

acceptance region

z/2-z/2

/2 /2

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A test is one-tailed when the alternate hypothesis, H1, states a direction.

H1: The mean yearly commissions earned by full-time realtors is more than $65,000. (µ>$65,000)

H1: The mean speed of trucks traveling on the 407 in Ontario is less than 120 kilometres per hour.

(µ<120)

H1: Less than 20 percent of the customers pay cash for their gasoline purchase. (p<.20)

ExamplesExamples

Tests of Significance

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5% Level of Significance =.05=.05

Reject Ho when z >1.65Reject Ho when z >1.65

0

= 5% rejection

region

1- = 95% acceptance

region

1.651.65

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A test is two-tailed when no direction is specified in the alternate hypothesis, H1

H1: The mean time Canadian families live in a particular home is not equal to 10

years. (µ10)

H1: The average speed of trucks travelling on the 407 in Ontario is different than 120 kph.

(µ120)

H1: The percentage of repeat customers within a week at Tim Horton’s is not 50%. (p .50)

ExamplesExamples

Tests of Significance

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5% Level of Significance

Reject Ho when z>1.96 or z< -1.96 Reject Ho when z>1.96 or z< -1.96

= 5% rejection

region

= 95% acceptance

region

0.025 0.025

1.96 & -1.96 are called “critical values”1.96 & -1.96 are called “critical values”

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Testing for the Population Mean: Large Sample,

Population Standard Deviation Known

Testing for the Population Mean: Large Sample,

Population Standard Deviation Known

Test Statistic to be used:

n/

Xz

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Testing for the Population Mean: Large Sample, Population Standard Deviation Known

Testing for the Population Mean: Large Sample, Population Standard Deviation Known

The processors of eye drop medication indicate on the label that the bottle contains 16 ml of medication.

The standard deviation of the process is 0.5 ml. A sample of 36 bottles from the last hour’s

production revealed a mean weight of 16.12 ml per bottle.

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At the .05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle

is different from 16 ml?

At the .05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle

is different from 16 ml?

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Hypothesis Test Hypothesis Test

State the null and alternate hypothesesState the null and alternate hypothesesStep 1Step 1


Identify the test statisticIdentify the test statisticStep 3Step 3

State the decision ruleState the decision ruleStep 4Step 4

Compute the test statistic and make a decision


Step 5Step 5

H0: µ = 16 H1: µ 16 = 0.05

Because we know the standard deviation, the test statistic is Z

Reject H0 if z > 1.96 or z < -1.96

44.1

365.0

00.1612.16

n

Xz

Do not reject the null hypothesis. We cannot conclude the mean is different from 16 ml.

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Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown

Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown

Rock’s Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the

mean monthly unpaid balance is more than $400.

Should Lisa conclude that the population mean is greater than $400, or is it reasonable to assume that the difference of $7 ($407-

$400) is due to chance?

A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation

to be $38.

The level of significance is set at .05. The level of significance is set at .05.

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When the sample is large, i.e. over 30, you can use the z-distribution as your test statistic.

Remember, use the best that you have!Remember, use the best that you have!

(Just replace the sample standard deviation for the population standard deviation)

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Step 5Step 5

H0: µ = 400 H1: µ > 400 = 0.05

Because the sample is large, we use the test statistic Z

Reject H0 if z > 1.645

42.2

n

Xz

Reject the hypothesis. H0 . Lisa can conclude that the mean unpaid balance is greater than

$400!

17238$

400$407$

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Testing for the Population Mean: Small Sample, Population Standard Deviation Unknown


ns

Xt

/

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The current production rate for producing 5 amp fuses at Ned’s Electric Co. is 250 per hour.



A new machine has been purchased and installed that, according to the supplier, will increase the production rate!

A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256 units,

with a sample standard deviation of 6 per hour.

At the .05 significance level, can Ned conclude that the new machine is

faster?

At the .05 significance level, can Ned conclude that the new machine is

faster?

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Step 5Step 5

H0: µ = 250 H1: µ > 250 = 0.05

Because the sample is small and is unknown, we use the t-test

Reject H0 if t > 1.833

162.3

n

Xt

Reject the hypothesis. H0 . Ned can conclude that the new machine will increase the

production rate!

106

250256

… 10 -1 = 9 degrees of freedom

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A P -Value is the probability, (assuming that the null hypothesis is true) of finding a value of the test statistic at least as extreme as the computed value

for the test!If the P-Value is smaller than the significance level,

H0 is rejected.

If the P-Value is larger than the significance level,

H0 is not rejected.

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Since P-value is smaller than of 0.05, reject H0.

The population mean is greater

than $400.

Since P-value is smaller than of 0.05, reject H0.

The population mean is greater

than $400.

Rock’s Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid

balance is more than $400. The level of significance is set at .05.

A random check of 172

unpaid balances revealed the sample mean to be $407

and the sample standard deviation to be $38.

Should Lisa conclude that the

population mean is greater than $400?

Rock’s Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid

balance is more than $400. The level of significance is set at .05.

A random check of 172

unpaid balances revealed the sample mean to be $407

and the sample standard deviation to be $38.

Should Lisa conclude that the

population mean is greater than $400?

= 0.05

P(z 2.42) =

Previously determined…

.5 - .4922 = .0078

42.2

ns

Xz

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The processors of eye drop medication indicate on the label that the bottle contains 16 ml of

medication. The standard deviation of the process is 0.5 ml. A sample of 36 bottles from last

hour’s production revealed a mean weight of 16.12 ml per bottle. At the .05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle is different from 16

ml?

= 0.05 = 0.05Previously determined…



= 2p(z |1.44|)= 2(.5 - .4251)= 2(.0749)= .1498

= 2p(z |1.44|)= 2(.5 - .4251)= 2(.0749)= .1498

Since .1498 > .05, do not reject H0.Since .1498 > .05, do not reject H0.

44.1

n

Xz

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Interpreting the Weight of Evidence against Ho

Interpreting the Weight of Evidence against Ho

If the P-value is less than …If the P-value is less than …

.10 we have some evidence that Ho is not true

.05 we have strong evidence that Ho is not true

.01 we have very strong evidence that Ho is not true

.001 we have extremely strong evidence that Ho is not true

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If the P-value is less than…If the P-value is less than….10 we have some evidence.05 we have strong evidence

.01 we have very strong evidence.001 we have extremely strong evidence

that Ho is not true

Since P-value is .0078Since P-value is .0078

… we have very strong

evidenceto conclude that the

population mean is greater than

$400!

… we have very strong

evidenceto conclude that the

population mean is greater than

$400!

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… is the fraction or percentage that indicates the

part of the population or sample having a

particular trait of interest

… is the fraction or percentage that indicates the

part of the population or sample having a

particular trait of interest

A Proportion

… is denoted by p … is found by:

Sample Proportion

sampledNumber

sample in the successes ofNumber p

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Testing a Single Population Proportion:

Testing a Single Population Proportion:


… is the symbol for sample proportion

… is the symbol for population proportionpp̂

p0 … represents a population proportion of interest

n

pp

ppz

)1(

ˆ

00

0

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In the past, 15% of the mail order solicitations for a certain charity

resulted in a financial contribution.

At the .05 significance level can it be concluded that the

new letter is more effective?

A new solicitation letter that has been drafted is sent to a sample of 200 people and

45 responded with a contribution.

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Step 5Step 5

= 0.05We will use the z-test

Reject the hypothesis. More than 15% are responding with a pledge, therefore, the new letter is more effective!

H1: p > .15

H0: p = .15

Reject H0 if z > 1.645

ppz ˆ

npp )1(ˆ

200)15.1(.15

20045 15.

97.2

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Relationship Between Hypothesis Testing Procedure and Confidence Interval

Estimation

Relationship Between Hypothesis Testing Procedure and Confidence Interval

Estimation

Case 1:Case 1:

Our decision rule can be restated as:

Do not reject H0 if 0 lies in the (1-) confidence

interval estimate of the population mean,

computed from the sample data

TEST

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0

=

rejection region

1- =

Confidence Interval region

Do not reject Ho when z falls in the confidence interval estimate

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Relationship Between Hypothesis Testing Procedure and

Confidence Interval Estimation



Case 2:Case 2: Lower-tailed test


Do not reject H0 if 0 is less than or equal to

the (1-) upper confidence bound for , computed from

the sample data.

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0

=

rejection region

1- =

confidence level region

Do not reject





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Case 3:Case 3: Upper-tailed test


Do not reject H0 if 0 is greater than or equal to

the (1-) lower confidence bound for , computed from the sample data.

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0

=

rejection region

1- = acceptance

region

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Level of Significance…is the probability of rejecting the null hypothesis

when it is actually true, i.e. Type I Error

…accepting the null hypothesis when it is actually false.

Type II Error

Type II ErrorType II Error

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Calculating the Probability of a Type II Error

Calculating the Probability of a Type II Error

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A batch of 5000 light bulbs either belong to a superior type, with a mean life of 2400

hours, or to an inferior type, with a mean life of 2000 hours. (By default,

the bulbs will be sold as the inferior type.)

A batch of 5000 light bulbs either belong to a superior type, with a mean life of 2400

hours, or to an inferior type, with a mean life of 2000 hours. (By default,

the bulbs will be sold as the inferior type.)

Suppose we select a sample of 4 bulbs. Find the probability of a

Type II error.

Suppose we select a sample of 4 bulbs. Find the probability of a

Type II error.

Both bulb distributions are normal, with a standard deviation of 300 hours. = 0.025.

Both bulb distributions are normal, with a standard deviation of 300 hours. = 0.025.

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H0: µ = 2000 H1: µ = 2400 = 0.025

As populations are normal, is known, we use the z-test

Reject H0 if the computed z > 1.96, or stated

another way,If the computed value x bar is greater than xu = 2000 +1.96(300/n), REJECT H0 in favour of H1

Superior: =2400 Inferior: =2000

=300 =0.025

Superior: =2400 Inferior: =2000

=300 =0.025

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Suppose H0 is false and H1 is true. i.e. the true value of µ is 2400,

then x bar is approximately normally distributed with a mean of

2400 and a standard deviation of /n = 300/n

Suppose H0 is false and H1 is true. i.e. the true value of µ is 2400,

then x bar is approximately normally distributed with a mean of

2400 and a standard deviation of /n = 300/n

…is the probability of not rejecting Ho

…is the probability that the value of x bar obtained will be less than or equal to xu

The probability of a Type II Error

Xu

X

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Suppose we select a sample of 4 bulbs. Then x bar has a mean of 2400 and a

sd of 300/4 = 150

Suppose we select a sample of 4 bulbs. Then x bar has a mean of 2400 and a

sd of 300/4 = 150

Xu = 2000+1.96(300/4) = 2294

A1 = 0.2611, giving us a left tail area of 0.24

A1 = 0.2611, giving us a left tail area of 0.24

70666.04300

24002294

n

Xz

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The probability of a Type II error is 0.24 i.e.=0.24The probability of a Type II error is 0.24 i.e.=0.24

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※ If we decrease the value of (alpha), the value z increases and the critical value xu

moves to the right, and therefore the value of (beta) increases.

Conversely, if we increase the value of (alpha), xu moves to the left, thereby

decreasing the value of (beta)

For a given value of (alpha), the value of (beta) can be decreased by increasing the sample size.

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Power of a TestPower of a Test

… is defined as the probability of rejecting H0 when H0 is false, or

…the probability of correctly identifying a true alternative hypothesis

…it is equal to (1-)

In previous example, = 0.24

Therefore, the test’s power is 1-0.24 = 0.76

In previous example, = 0.24

Therefore, the test’s power is 1-0.24 = 0.76

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Test your learning…Test your learning…

www.mcgrawhill.ca/college/lindClick on…Click on…

Online Learning Centrefor quizzes

extra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat data…and much more!

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This completes Chapter 10This completes Chapter 10

copyright © 2003 mcgraw-hill ryerson limited. all rights reserved. 10 - 1

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