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Chapter 8Introduction to Hypothesis Testing

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Name of the game… Hypothesis testing

Statistical method that uses sample data to evaluate a hypothesis about a population

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Experimental Hypotheses

Experimental hypotheses describe the

predicted outcome we may or may not find in

an experiment.

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Order of Procedure State hyp about pop Before selecting a sample..use hyp to

predict characteristics that sample should have

Obtain random sample Compare sample data w/ prediction made

in hyp

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Of interest to the researcher… Did the treatment have any effect on the

individuals Must be large (significant) differences in

means

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New Statistical Notation

The symbol for greater than is >.

The symbol for less than is <.

The symbol for greater than or equal tois ≥.

The symbol for less than or equal tois ≤.

The symbol for not equal to is ≠.

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The Role of InferentialStatistics in Research

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Sampling Error

Remember:

Sampling error results when random chance

produces a sample statistic that does not

equal the population parameter it represents.

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Setting up Inferential Procedures

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Null Hypothesis H0

The null hypothesis describes the

population parameters that the sample data

represent if the predicted relationship does

not exist.

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Alternative Hypothesis H1

The alternative hypothesis describes the

population parameters that the sample data

represent if the predicted relationship exists.

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A Graph Showing the Existence of a Relationship

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A Graph Showing That a Relationship Does Not Exist

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Interpreting Significant Results When we reject H0 and accept H1, we do

not prove that H0 is false

While it is unlikely for a mean that lies within the rejection region to occur, the sampling distribution shows that such means do occur once in a while

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Failing to Reject H0

When the statistic does not fall beyond the critical value, the statistic does not lie within the region of rejection, so we do not reject H0

When we fail to reject H0 we say the results are nonsignificant. Nonsignificant indicates that the results are likely to occur if the predicted relationship does not exist in the population.

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Interpreting Nonsignificant Results When we fail to reject H0, we do not prove

that H0 is true

Nonsignificant results provide no convincing evidence—one way or the other—as to whether a relationship exists in nature

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Errors in StatisticalDecision Making

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Type I Errors A Type I error is defined as rejecting H0 when H0

is true

In a Type I error, there is so much sampling error that we conclude that the predicted relationship exists when it really does not

The theoretical probability of a Type I error equals

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Alpha Probability that the test will lead to a Type I

error Alpha level determines the probability of

obtaining sample data in the critical region even though the null hypo is true

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Type II Errors A Type II error is defined as retaining H0 when H0

is false (and H1 is true)

In a Type II error, the sample mean is so close to the described by H0 that we conclude that the predicted relationship does not exist when it really does

The probability of a Type II error is

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Power The goal of research is to reject H0 when H0

is false

The probability of rejecting H0 when it is false is called power

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Possible Results of Rejecting or Retaining H0

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Parametric Statistics Parametric statistics are procedures that require

certain assumptions about the characteristics of the populations being represented. Two assumptions are common to all parametric procedures: The population of dependent scores forms a normal

distribution

and The scores are interval or ratio.

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Nonparametric Procedures

Nonparametric statistics are inferential

procedures that do not require stringent

assumptions about the populations being

represented.

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Robust Procedures

Parametric procedures are robust. If the data

don’t meet the assumptions of the procedure

perfectly, we will have only a negligible

amount of error in the inferences we draw.

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Predicting a Relationship A two-tailed test is used when we predict

that there is a relationship, but do not predict the direction in which scores will change.

A one-tailed test is used when we predict the direction in which scores will change.

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The One-Tailed Test

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One-Tailed Hypotheses In a one-tailed test, if it is hypothesized that the independent

variable causes an increase in scores, then the null hypothesis is that the population mean is less than or equal to a given value and the alternative hypothesis is that the population mean is greater than the same value. For example: H0: ≤ 50 Ha: > 50

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A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of

Whether Scores Increase

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One-Tailed Hypotheses In a one-tailed test, if it is hypothesized that the independent

variable causes a decrease in scores, then the null hypothesis is that the population mean is greater than or equal to a given value and the alternative hypothesis is that the population mean is less than the same value. For example: H0: ≥ 50 Ha: < 50

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A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores

Decrease

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Choosing One-Tailed Versus Two-Tailed Tests

Use a one-tailed test only when confident of

the direction in which the dependent variable

scores will change. When in doubt, use a

two-tailed test.

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Performing the z-Test

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The z-Test

The z-test is the procedure for computing a

z-score for a sample mean on the sampling

distribution of means.

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Assumptions of the z-Test1. We have randomly selected one sample

2. The dependent variable is at least approximately normally distributed in the population and involves an interval or ratio scale

3. We know the mean of the population of raw scores under some other condition of the independent variable

)( X4. We know the true standard deviation of the population

described by the null hypothesis

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Setting up for a Two-Tailed Test1. Choose alpha. Common values are 0.05

and 0.01.

2. Locate the region of rejection. For a two-tailed test, this will involve defining an area in both tails of the sampling distribution.

3. Determine the critical value. Using the chosen alpha, find the zcrit value that gives the appropriate region of rejection.

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A Sampling Distribution for H0 Showing the Region of Rejection for = 0.05 in a Two-tailed Test

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In a two-tailed test, the null hypothesis states that the population mean equals a given value. For example, H0: = 100.

In a two-tailed test, the alternative hypothesis states that the population mean does not equal the same given value as in the null hypothesis. For example, Ha: 100.

Two-Tailed Hypotheses

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Computing z

X

Xz

obt

NX

X

• The z-score is computed using the same formula as before

where

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Rejecting H0

When the zobt falls beyond the critical value, the statistic lies in the region of rejection, so we reject H0 and accept Ha

When we reject H0 and accept Ha we say the results are significant. Significant indicates that the results are too unlikely to occur if the predicted relationship does not exist in the population.

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Interpreting Significant Results When we reject H0 and accept Ha, we do

not prove that H0 is false

While it is unlikely for a mean that lies within the rejection region to occur, the sampling distribution shows that such means do occur once in a while

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Failing to Reject H0

When the zobt does not fall beyond the critical value, the statistic does not lie within the region of rejection, so we do not reject H0

When we fail to reject H0 we say the results are nonsignificant. Nonsignificant indicates that the results are likely to occur if the predicted relationship does not exist in the population.

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Interpreting Nonsignificant Results When we fail to reject H0, we do not prove

that H0 is true

Nonsignificant results provide no convincing evidence—one way or the other—as to whether a relationship exists in nature

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1. Determine the experimental hypotheses and create the statistical hypothesis

Summary of the z-Test

X2.Compute and compute zobt

3.Set up the sampling distribution

4.Compare zobt to zcrit