Chapter 8Chapter 8

Introduction to Hypothesis TestingIntroduction to Hypothesis Testing

Hypothesis TestingHypothesis Testing

Hypothesis testing is a statistical Hypothesis testing is a statistical procedureprocedure

Allows researchers to use sample data to Allows researchers to use sample data to draw inferences about the population of draw inferences about the population of interestinterest

Although the details of a hypothesis test Although the details of a hypothesis test will change from one situation to another, will change from one situation to another, the general process will remain constantthe general process will remain constant

Hypothesis Testing (cont.)Hypothesis Testing (cont.)

For this chapter, we have to understand z-For this chapter, we have to understand z-scores, probability, and the distribution of scores, probability, and the distribution of sample means to create a new statistical sample means to create a new statistical procedure known as a hypothesis test.procedure known as a hypothesis test.

Hypothesis TestHypothesis Test

A hypothesis test is a statistical method that A hypothesis test is a statistical method that uses sample data to evaluate a hypothesis uses sample data to evaluate a hypothesis about a population mean.about a population mean.

Underlying logicUnderlying logic State a hypothesis about a populationState a hypothesis about a population Use the hypothesis to predict the characteristics that Use the hypothesis to predict the characteristics that

the sample should havethe sample should have Obtain a random sample from the populationObtain a random sample from the population Compare the obtained sample data with the prediction Compare the obtained sample data with the prediction

that was made from the hypothesisthat was made from the hypothesis

A hypothesis test is typically used in the A hypothesis test is typically used in the context of a research studycontext of a research studyOnce a researcher completes a research Once a researcher completes a research study, a hypothesis test is used to study, a hypothesis test is used to evaluate the resultsevaluate the results Details of the hypothesis test will change from Details of the hypothesis test will change from

one situation to anotherone situation to another

For now, we will focus on the most For now, we will focus on the most common hypothesis testscommon hypothesis tests

SituationSituation: A researcher is using one sample : A researcher is using one sample to examine to examine oneone unknown population unknown population

The purpose of the research is to The purpose of the research is to determine the effect of the treatment on determine the effect of the treatment on the individuals in the population.the individuals in the population.

The goal is to determine what happens to The goal is to determine what happens to the population after the treatment is the population after the treatment is administered.administered.

Figure 8.1Figure 8.1

The research situation of hypothesis testingThe research situation of hypothesis testing

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Begin with a known population (before the treatment)

Assume the population forms a normal distribution

Purpose is to determine the effect of the treatment on the individuals in the population

What happens to the population after the treatment is administered?

The problem is to determine whether or The problem is to determine whether or not the treatment has an effect;not the treatment has an effect;The parameters are known for the The parameters are known for the population before treatment;population before treatment;The question is whether or not the The question is whether or not the population mean is changed by the population mean is changed by the treatment;treatment;To help answer the question, the To help answer the question, the researcher obtains a sample of individuals researcher obtains a sample of individuals who have received the treatment.who have received the treatment.

The basic research situation for hypothesis testingThe basic research situation for hypothesis testing

To simplify the hypothesis-testing To simplify the hypothesis-testing situation, one basic assumption is made situation, one basic assumption is made about the effect of the treatmentabout the effect of the treatment

If the treatment has any effect, it is simply If the treatment has any effect, it is simply to add a constant amount to (or subtract a to add a constant amount to (or subtract a constant amount from) each individual’s constant amount from) each individual’s score.score.

Remember a constant will not change the Remember a constant will not change the shape of the population, nor will it change shape of the population, nor will it change the standard deviationthe standard deviation

The population after the treatment will also The population after the treatment will also have the same shape as the original have the same shape as the original population and the same s.d.population and the same s.d.

The sample in the research studyThe sample in the research study

The goal of the hypothesis test is to The goal of the hypothesis test is to determine whether or not the treatment determine whether or not the treatment has any effect on the individuals in the has any effect on the individuals in the populationpopulationBecause the populations are usually too Because the populations are usually too big, we use a sample.big, we use a sample.The hypothesis test will use the sample to The hypothesis test will use the sample to test a hypothesis about the unknown test a hypothesis about the unknown population mean.population mean.

Because a hypothesis test is a formalized Because a hypothesis test is a formalized procedure that follows a standard series of procedure that follows a standard series of operations,operations,Researchers have a standardized method Researchers have a standardized method for evaluating the results of their research for evaluating the results of their research studies;studies;Other researchers will understand how the Other researchers will understand how the data were evaluated and how conclusions data were evaluated and how conclusions were reached.were reached.

Hypothesis test formal structureHypothesis test formal structure

Will use a four-step process Will use a four-step process

Will be used throughout the rest of the Will be used throughout the rest of the bookbook

Example 8.1Example 8.1

Psychologists note that stimulation during Psychologists note that stimulation during infancy can have profound effects on the infancy can have profound effects on the development of infant rats.development of infant rats.

Based on data, one might theorize that Based on data, one might theorize that increased stimulation early in life can be increased stimulation early in life can be beneficial.beneficial.

Could this theory be applied to infants?Could this theory be applied to infants?

Mean weight of 2-year olds is Mean weight of 2-year olds is = 26 lbs.= 26 lbs.

With a With a = 4 lbs= 4 lbs

n=16n=16

Sample parents given instructions for Sample parents given instructions for working with their infantsworking with their infants

At age 2, will weigh the childrenAt age 2, will weigh the children

We do not know what will happen to the We do not know what will happen to the mean weight for 2-year old childrenmean weight for 2-year old children

Do have a sample of 16 infants that we Do have a sample of 16 infants that we can be sure about.can be sure about.

Can use this sample to draw inferences Can use this sample to draw inferences about the unknown populationabout the unknown population

Follow the four stepsFollow the four steps

StepsSteps

1. State the hypothesis1. State the hypothesis

2. Set the criteria for a decision2. Set the criteria for a decision

3. Collect data and compute sample 3. Collect data and compute sample statisticsstatistics

4. Make a decision4. Make a decision

Four StepsFour StepsStep 1Step 1State the hypothesesState the hypotheses Actually state two hypothesesActually state two hypotheses Both in terms of population parametersBoth in terms of population parameters

Null hypothesesNull hypotheses States that the treatment has no effect.States that the treatment has no effect. Identified by the symbol HIdentified by the symbol Hoo

H stands for hypothesisH stands for hypothesisO indicates that this is the zero-effectO indicates that this is the zero-effect

HHoo= = infants handled = 26 poundsinfants handled = 26 pounds

Step 1Step 1

The second hypothesis is the opposite of The second hypothesis is the opposite of the null hypothesisthe null hypothesis

Called the scientific or alternative Called the scientific or alternative hypothesis (Hhypothesis (H11))

States that the treatment has an effect on States that the treatment has an effect on the dependent variablethe dependent variable

HH11= = infants handled >< = 26 poundsinfants handled >< = 26 pounds

An alternative hypothesis simply states that An alternative hypothesis simply states that there will be some type of changethere will be some type of changeIt might be necessary to specify the direction of It might be necessary to specify the direction of the effect in Hthe effect in H1 1 > 26 pounds> 26 pounds

This is called directional hypothesis testThis is called directional hypothesis testNote that both hypotheses refer to a population Note that both hypotheses refer to a population whose mean is unknownwhose mean is unknown The population of infants who receive extra handling The population of infants who receive extra handling

early in lifeearly in life

Step 2Step 2

Set the Criteria for a DecisionSet the Criteria for a Decision

Will eventually use the data from the sample to Will eventually use the data from the sample to evaluate the credibility of the null hypothesisevaluate the credibility of the null hypothesis

Will use the null hypothesis to predict the kind of Will use the null hypothesis to predict the kind of sample mean that ought to be obtainedsample mean that ought to be obtained

We will determine exactly what sample means We will determine exactly what sample means are consistent with the null hypothesis and what are consistent with the null hypothesis and what sample means are at odds with the null sample means are at odds with the null hypothesishypothesis

Begin by examining all the possible Begin by examining all the possible sample means that could be obtained in sample means that could be obtained in the null hypothesis is truethe null hypothesis is true

Distribution of sample means should be Distribution of sample means should be centered at centered at = 26= 26

The distribution of sample means is then The distribution of sample means is then divided into two sections.divided into two sections.

1. Sample means that are likely to be 1. Sample means that are likely to be obtained if Ho is trueobtained if Ho is true Those close to the null hypothesisThose close to the null hypothesis

2. Sample means that are very unlikely to 2. Sample means that are very unlikely to be obtained if Ho is truebe obtained if Ho is true Those that are very different from the null Those that are very different from the null

hypothesishypothesis

The High probability samples are located in the The High probability samples are located in the center of the distribution and have sample center of the distribution and have sample means close to the value specified in the null means close to the value specified in the null hypothesis.hypothesis.The low-probability samples are located in the The low-probability samples are located in the extreme tails of the distribution.extreme tails of the distribution.After the distribution has been divided in this After the distribution has been divided in this way, we can compare our sample data with the way, we can compare our sample data with the values in the distributionvalues in the distributionWe can determine whether our sample mean is We can determine whether our sample mean is consistent with the null hypothesisconsistent with the null hypothesis

Figure 8.2Figure 8.2

The set of potential samples is divided into The set of potential samples is divided into those that are likely to be obtained and those that are likely to be obtained and those that are very unlikely if the null those that are very unlikely if the null hypothesis is true.hypothesis is true.

Figure 8.2Figure 8.2

The distribution of sample means if the null hypothesis is trueThe distribution of sample means if the null hypothesis is true

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Alpha LevelAlpha Level

To find the boundaries that separate the To find the boundaries that separate the high-probability samples from the low-high-probability samples from the low-probability samples, we must define probability samples, we must define exactly what is meant by “low” probability exactly what is meant by “low” probability and “high” probability.and “high” probability.This is accomplished by selecting a This is accomplished by selecting a specific probability value, which is known specific probability value, which is known as the as the level of significancelevel of significance or the or the alpha alpha level level for the hypothesis test.

The alpha (The alpha () value is a small probability ) value is a small probability that is used to identify the low-probability that is used to identify the low-probability samples.samples. With a With a we will separate the we will separate the most likely 95% of the sample means (the most likely 95% of the sample means (the central values)central values)

Extreme ValuesExtreme Values

The extremely unlikely values, as defined The extremely unlikely values, as defined by the alpha level, make up what is called by the alpha level, make up what is called the the critical regioncritical regionExtreme values are inconsistent with the null hypothesisIf data produce a sample mean that is located in the critical region, we will conclude that the data are inconsistent with the null hypothesis

Technically, the critical region is defined Technically, the critical region is defined by sample outcomes that are very unlikely by sample outcomes that are very unlikely to occur if the treatment has no effectto occur if the treatment has no effect

That is, if the null hypothesis is trueThat is, if the null hypothesis is true

It is almost impossible if there is no It is almost impossible if there is no treatment effecttreatment effect

The boundaries for the critical regionThe boundaries for the critical region

To determine the exact location for the To determine the exact location for the boundaries that define the critical regionboundaries that define the critical region Use the alpha-level probabilityUse the alpha-level probability Unit normal tableUnit normal table

Find the boundaries that separate the extreme Find the boundaries that separate the extreme 5% from the middle 95%5% from the middle 95%Split the 5%Split the 5% 2.5% (or 0.0250) in each tail2.5% (or 0.0250) in each tail

Z = +/- 1.96Z = +/- 1.96Thus, for any normal distribution, the extreme Thus, for any normal distribution, the extreme 5% is in the tails of the distribution beyond z = 5% is in the tails of the distribution beyond z = 1.96 and z = -1.961.96 and z = -1.96The values define the boundaries of the critical The values define the boundaries of the critical region for a hypothesis test using region for a hypothesis test using

Figure 8.3Figure 8.3

The critical region for an alpha of .05The critical region for an alpha of .05

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1% or .0100 is split between the two tails1% or .0100 is split between the two tails

The proportion in each tail is .0050The proportion in each tail is .0050

z= +/- 2.58z= +/- 2.58

.01% or .0010 is split between the two tails.01% or .0010 is split between the two tails

The proportion in each tail is .0005The proportion in each tail is .0005

z= +/- 3.30z= +/- 3.30

Collect Data and Compute Sample StatisticsCollect Data and Compute Sample Statistics

Step 3Step 3

Collect sample dataCollect sample data

Collect the data after the sample has been Collect the data after the sample has been selectedselected Assures an honest objective evaluation of dataAssures an honest objective evaluation of data

Raw data are summarized with the appropriate Raw data are summarized with the appropriate statisticsstatistics Compute the sample mean (in this example)Compute the sample mean (in this example) Compare the sample mean with the null hypothesisCompare the sample mean with the null hypothesis

To compare the sample mean with the null To compare the sample mean with the null hypothesis, compute a z-score that describes hypothesis, compute a z-score that describes exactly where the sample mean is located relative exactly where the sample mean is located relative to the hypothesized population mean from Hto the hypothesized population mean from Ho o

Z = M – Z = M – MM

M = sample meanM = sample mean

z = sample mean – hypothesized population mean

Standard error between M and

Make a DecisionMake a Decision

Use the z-score value obtained in Step 3 Use the z-score value obtained in Step 3 to make a decision about the null to make a decision about the null hypothesis according to the criteria hypothesis according to the criteria established in Step 2established in Step 2

Two possible decisionsTwo possible decisions Accept the null hypothesisAccept the null hypothesis Reject the null hypothesisReject the null hypothesis

Sample data fall into critical regionSample data fall into critical region

Rejecting the Null Hypothesis vs. Rejecting the Null Hypothesis vs. Proving the Alternative HypothesisProving the Alternative Hypothesis

The reason for focusing on the null The reason for focusing on the null hypothesis as compared to the alternative hypothesis as compared to the alternative hypothesis comes from the limitations of hypothesis comes from the limitations of inferential logicinferential logic

Remember that we want to use the Remember that we want to use the sample data to draw conclusions, or sample data to draw conclusions, or inferences, about a populationinferences, about a population

Logically, it is easier to demonstrate that a Logically, it is easier to demonstrate that a universal (population) hypothesis is false universal (population) hypothesis is false than to demonstrate that it is truethan to demonstrate that it is trueIt would be difficult to state “the treatment It would be difficult to state “the treatment has an effect” as the hypothesis and then has an effect” as the hypothesis and then try to prove that this is truetry to prove that this is trueTherefore, we state the null hypothesis Therefore, we state the null hypothesis “the treatment has no effect” and try to “the treatment has no effect” and try to show that it is falseshow that it is false

In the end, we still demonstrate that the In the end, we still demonstrate that the treatment does have an effect.treatment does have an effect.

We find support for the alternative We find support for the alternative hypothesis by disproving (rejecting) the hypothesis by disproving (rejecting) the null hypothesisnull hypothesis