cbesta2chap09slides

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NOTES

ELGA 1: Effective communicators

Know and be familiar with the correctnomenclature and symbols for parameters and

statistics

ELGA 2: Critical and creative thinkers

Recognize problems as applications of thevarious tests of hypothesis which have beendiscussed

Analyze and solve statistical problems andinterpret statistical results

Chapter 9

Tests of Hypothesis

ELGA 3: Technically proficient and competentprofessionals and leaders

Learn to use the calculator and the computer ingenerating statistical results

Be familiar with the use of basic software(Microsoft Excel and its add-in PHSTAT) in statisticalanalysis and decision-making;

ELGA 4: Service-driven, ethical, and sociallyresponsible citizens

Realize the significance of statistics in

business decision-making of entrepreneurs andcorporate managers and how statistics can

contribute to development

Chapter 9Tests of Hypothesis


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Chapter 9

Hypothesis Testing

Chapter Outline

Developing Null and AlternativeHypotheses

Type I and Type II Errors

One-Tailed Tests About a PopulationMean: Large-Sample Case

Two-Tailed Tests About a PopulationMean: Large-Sample Case

Tests About a Population Mean: Small-Sample Case

continued

Chapter 9

Hypothesis Testing

Tests About a Population Proportion

Hypothesis Testing and Decision Making

Calculating the Probability of Type I & IIErrors

What are Hypotheses?

Research Hypothesis a statement of what the researcher believes will

be the outcome of an experiment or a study.

Statistical Hypotheses a more formal structure derived from the research

hypothesis. A conjecture regarding one or several

population parameters. Substantive Hypotheses

a statistically significant difference does not implyor mean a material, substantive difference.


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Developing Null andAlternative Hypotheses

Definition. The null hypothesis, denoted by H0is the conjecture that is held as true unless

there is sufficient evidence to concludeotherwise. The null hypothesis always containsan equality and hence, in terms of populationparameters, comes in the form H0: = 0 orH0: 0, or H0: 0. It is the Status Quo.Manufacturers claims are usually given thebenefit of the doubt and stated as the nullhypothesis.

Definition. The alternative hypothesisdenoted by HA is the conjecture that covers all

situations not covered by the null hypothesis.The alternative hypothesis never contains an

equality and hence, in terms of populations

parameters, comes in the form HA: 0 or HA: < 0, or HA: > 0. The burden of proof fallson the alternative hypothesis.

Developing Null and

Alternative Hypotheses

Test Statistic

Definition. The test statistic is a measure ofhow close the sample results have come to thehypothesized value of the parameter in the nullhypothesis. The test statistic follows a well-known distribution such as Z, t, binomial, etc.


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Decision Rule

Definition. The decision rule is a rule thatspecifies under what conditions the nullhypothesis may be rejected.

There are two decisions that can be made in atest of hypothesis, namely, to reject it, or not toreject (accept) it.

Example: Accepting a shipment of goodsfrom a supplier or returning the shipment ofgoods to the supplier.

Errors in Hypothesis Testing

Hypothesis testing is not a foolproof

procedure. Remember that statisticians basetheir decisions on sample results, andsometimes, sample results could lead themto make incorrect decisions.

Types of Errors

There are two types of errors that can be made when

carrying out a test of hypothesis.

A Type I error occurs when one rejects the nullhypothesis when in fact it is true. The probability of

committing a Type I error is denoted by and it iscalled the level of significance.

A Type II error occurs when one fails to reject thenull hypothesis when in fact it is false. The probabilityof committing a Type II error is denoted by .


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Types of Errors

Type II errorCorrect DecisionDont Reject H0

Correct DecisionType I errorReject H0

H0 FalseH0 TrueDecision

Actual Situation

Critical Region

Definition. The critical region, or rejection region,is a range of numbers such that if the value of thetest statistic falls in this range, it will lead to therejection of the null hypothesis.

The critical numbers determines the critical region.The rejection region is designed so that, before thesampling takes place, our test statistic will have a

probability of of falling within the rejection region ifthe null hypothesis is true.

Region of Non-Rejection

Definition. The region of non-rejection is therange of values (also determined by the criticalnumbers) that will lead us not to reject the nullhypothesis if the test statistic should fall withinthis region. The region or non-rejection is

designed so that before the sampling takesplace, our test statistic will have a probability of1 - of falling in this region if the nullhypothesis is true.


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00>>00 z/2

Two-Tailed Tests about a Population

Mean: Large-Sample Case (n> 30)

zx

n=

0

/z

x

n=

0

/z

x

s n=

0/

zx

s n=

0/

=40 oz

Non Rejection Region

Rejection Region

Critical Value

Rejection Region

Critical Value

=40 oz


Rejection Region

Critical Value

Rejection Region

Critical Value

00>>00


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Hypotheses

H0: =0Ha: 0

Test Statistic Unknown, normal population

Rejection Rule Reject H0 if |t| > t/2

Two-Tailed Tests about a PopulationMean: Small-Sample Case (n< 30)

0

/xts n

= 0/

xt

s n

==40 oz


Rejection Region

Critical Value

Rejection Region

Critical Value

=40 oz


Rejection Region

Critical Value

Rejection Region

Critical Value

00 >> 00 z/2

Two-Tailed Tests about aPopulation Proportion

0

0 0(1 )

p p

p p

n

z

= 0

0 0(1 )

p p

p p

n

z

=

=40 oz


Rejection Region

Critical Value

Rejection Region

Critical Value

=40 oz


Rejection Region

Critical Value

Rejection Region

Critical Value


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Example: Metro EMS Null and Alternative Hypotheses

A major west coast city provides one of themost comprehensive emergency medical

services in the world. Operating in a multiplehospital system with approximately 20 mobilemedical units, the service goal is to respond tomedical emergencies with a mean time of 12minutes or less.

The director of medical services wants toformulate a hypothesis test that could use asample of emergency response times todetermine whether or not the service goal of12 minutes or less is being achieved.

Example: Arnolds Diner

A soft-drink machine at Arnolds Diner is

regulated so that the amount of drinkdispensed is approximately normallydistributed with a mean of 200 ml. and astandard deviation of 15 ml. The machine is

checked periodically by taking a sample of 9

drinks and computing the mean content.Based on the sample results, a decision is tobe made whether the machine is operatingsatisfactorily, or needs corrective action.

State the null and alternative hypotheses.What kind of test is this?

Example: Coffee

A producer of a certain brand of coffee claimsthat at least 20% of all coffee drinkers preferits product to the major competing brand.


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The 8-step procedure

!"#!$

"!!%%!!&$'( )#*! *!+ !%$%"", "- "!#"$!. !'#!/ !"#! 0'$"!

The equality part of the hypotheses alwaysappears in the null hypothesis.

In general, a hypothesis test about the value

of a population mean must take one of thefollowing three forms (where0 is thehypothesized value of the population mean).

H0: >0 H0:


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Example: Sewer Pipe

1!#!2#'0$$!%& 344!#%!!'#%"#0!''!%-4%!#'0$$!%!,.4!#%!!!#!%!#$ 344-!%$%""

The Use of p-Values

The p-value is the probability of obtaining asample result that is at least as unlikely aswhat is observed.

The p-value can be used to make thedecision in a hypothesis test by noting that:

if the p-value is less than the level ofsignificance , the value of the test statisticis in the rejection region.

if the p-value is greater than or equal to ,the value of the test statistic is not in therejection region.

Reject H0 if the p-value < .

P-Value for Normality Cases

In general, for a one tailed test concerning normallydistributed test statistics,

(a) if the HA contains a , the p-value is the areaP(Z > z) where z is the value of the test statistic.

(c) if the HA contains a , the p-value is 2P(Z > z )where z is the value of the test statistic.


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Example: Glow Toothpaste Two-Tailed Tests about a Population Mean:

Large n

The production line for Glow toothpaste is

designed to fill tubes of toothpaste with a meanweight of 6 ounces.

Periodically, a sample of 30 tubes will beselected in order to check the filling process.Quality assurance procedures call for the

continuation of the filling process if the sampleresults are consistent with the assumption thatthe mean filling weight for the population oftoothpaste tubes is 6 ounces; otherwise the

filling process will be stopped and adjusted.

Example: Glow Toothpaste Two-Tailed Test about a Population Mean:

Large n

A hypothesis test about the population mean can

be used to help determine when the fillingprocess should continue operating and when itshould be stopped and corrected.

Assume that a sample of 30 toothpaste tubes

provides a sample mean of 6.1 ounces andstandard deviation of 0.2 ounces. Use a 0.05

level of significance.

TwoTwo--Tailed Test about a Population Mean:Tailed Test about a Population Mean:

LargeLarge nn

A hypothesis test about the population mean canA hypothesis test about the population mean can

be used to help determine when the fillingbe used to help determine when the filling

process should continue operating and when itprocess should continue operating and when it

should be stopped and corrected.should be stopped and corrected.

Assume that a sample of 30 toothpaste tubesAssume that a sample of 30 toothpaste tubes

provides a sample mean of 6.1 ounces andprovides a sample mean of 6.1 ounces and

standard deviation of 0.2 ounces. Use a 0.05standard deviation of 0.2 ounces. Use a 0.05

level of significance.level of significance.

Confidence Interval Approach to aTwo-Tailed Test about a PopulationMean Select a simple random sample from the

population and use the value of the samplemean to develop the confidence interval for

the population mean.

If the confidence interval contains the

hypothesized value0, do not reject H0.Otherwise, reject H0.

xx


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Test Statistic Unknown

This test statistic has a tdistribution with n- 1 degrees offreedom.

Rejection Rule One-Tailed Two-Tailed

H0: t

H0: >0 Reject H0 if t< -t

H0: =0 Reject H0 if |t| > t/2

Tests about a Population Mean:Small-Sample Case (n< 30)

tx

s n=

0

/t

x

s n=

0

/

Example: Highway Patrol One-Tailed Test about a Population Mean: Small n

A State Highway Patrol periodically samplesvehicle speeds at various locations on a particularroadway. The sample of vehicle speeds is used to testthe hypothesis

H0: < 65.

The locations where H0 is rejected are deemed the bestlocations for radar traps.

At Location F, a sample of 16 vehicles shows amean speed of 68.7375 mph with a standard deviation

of 2.935 mph. Use an = .05 to test the hypothesis.

p-Values and the tDistribution

The format of the tdistribution table providedin most statistics textbooks does not havesufficient detail to determine the exact p-

value for a hypothesis test.

However, we can still use the tdistribution

table to identify a range for the p-value. An advantage of computer software

packages is that the computer output will

provide the p-value for the

tdistribution.


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Example: Kelloggs

According to Dietary Goals for the United States, high

sodium intake may be related to ulcers, stomachcancer, and migraine headaches. The humanrequirement for salt is only 220 milligrams per day,which is surpassed in most single servings of ready-to-eat cereals. If a random sample of 20 similarservings of Special K has a mean sodium content of244 milligrams of sodium and a standard deviation of24.5 milligrams, does this suggest at the 0.05 level ofsignificance that the average sodium content forsingle servings of Special K is greater than 220milligrams? Assume the distribution of sodiumcontents to be normal.

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The equality part of the hypotheses always appears inthe null hypothesis.

In general, a hypothesis test about the value of apopulation proportion pmust take one of the followingthree forms (where p0 is the hypothesized value of thepopulation proportion).

H0: p> p0 H0: p< p0 H0: p= p0Ha: p< p0 Ha: p> p0 Ha: p p0

Summary of Forms for Null and AlternativeHypotheses about a Population Proportion


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Test Statistic

where:

Rejection Rule

One-Tailed Two-Tailed

H0: p< p0 Reject H0 if z > zH0: p> p0 Reject H0 if z < -zH0: p= p0 Reject H0 if |z| > z/2

Tests about a Population Proportion:Large-Sample Case (np> 5 and n(1 - p) > 5)

zp p

p

= 0

zp p

p

= 0

p p pn

= 0 01( )p p pn

= 0 01( )

Example: NSC Two-Tailed Test about a Population Proportion: Large n

For a Christmas and New Years week, theNational Safety Council estimated that 500 peoplewould be killed and 25,000 injured on the nationsroads. The NSC claimed that 50% of the accidentswould be caused by drunk driving.

A sample of 120 accidents showed that 67 werecaused by drunk driving. Use these data to test the

NSCs claim with = 0.05.

Example: Drug Screening

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Computing Type I error

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End of Chapter 9

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