Chapter 9: Testing Hypotheses Overview Research and null hypotheses One and two-tailed tests Type I...

Chapter 9: Testing Hypotheses

• Overview• Research and null hypotheses• One and two-tailed tests• Type I and II Errors• Testing the difference between two

means• t tests

© 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Is one group scoring significantly higher on average than another group?

Is a group statistically different from another on a particular dimension?

Is Group A’s mean higher than Group B’s?

Do people living in rural communities live longer than those in urban or suburban areas?

Do students from private high schools perform better in college than those from public high schools?

Is the average number of years with an employer lower or higher for large firms (over 100 employees) compared to those with fewer than 100 employees?

General Examples of Hypothesis Testing

Specific Examples of Hypothesis Testing

• Statistical hypothesis testing – A procedure that allows us to evaluate hypotheses about population parameters based on sample statistics.

• Research hypothesis (H1) – A statement reflecting the substantive hypothesis. It is always expressed in terms of population parameters, but its specific form varies from test to test.

• Null hypothesis (H0) – A statement of “no difference,” which contradicts the research hypothesis and is always expressed in terms of population parameters.

Testing Hypotheses


Research and Null Hypotheses

One Tail — specifies the hypothesized direction• Research Hypothesis:

H1: 12, or 12> 0• Null Hypothesis:

H0: 12, or 12= 0

Two Tail — direction is not specified (more common)

• Research Hypothesis: H1: 12, or 12= 0

• Null Hypothesis: H0: 12, or 12= 0


One-Tailed Tests

One-tailed hypothesis test – A hypothesis test in which the alternative is stated in such a way that the probability of making a Type I error is entirely in one tail of a sampling distribution.

Right-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the right tail of the sampling distribution.

Left-tailed test – A one-tailed test in which the sample outcome is hypothesized to be at the left tail of the sampling distribution.


Two-Tailed Tests

Two-tailed hypothesis test – A hypothesis test in which the region of rejection falls equally within both tails of the sampling distribution.


Probability Values

Z statistic (obtained) – The test statistic computed by converting a sample statistic (such as the mean) to a Z score. The formula for obtaining Z varies from test to test.

P value – The probability associated with the obtained value of Z.


Probability Values


Probability Values

Alpha ( ) – The level of probability at which the null hypothesis is rejected. It is customary to set alpha at the .05, .01, or .001 level.


Five Steps to Hypothesis Testing

(1) Making assumptions(2) Stating the research and null

hypotheses and selecting alpha(3) Selecting the sampling

distribution and specifying the test statistic

(4) Computing the test statistic(5) Making a decision and interpreting

the results


• Type I error (false rejection error)the probability (equal to ) associated with rejecting a true null hypothesis.

• Type II error (false acceptance error)the probability associated with failing to reject a false null hypothesis.

Based on sample results, the decision made is to…

reject H0 do not reject H0

In the true Type I correct population error () decisionH0 is ...

false correct Type II error decision

Type I and Type II Errors


t-Test

t statistic (obtained) – The test statistic computed to test the null hypothesis about a population mean when the population standard deviation is unknown and is estimated using the sample standard deviation.

t distribution – A family of curves, each determined by its degrees of freedom (df). It is used when the population standard deviation is unknown and the standard error is estimated from the sample standard deviation.

Degrees of freedom (df) – The number of scores that are free to vary in calculating a statistic.


t Distribution


t Distribution Table

t-Test for Difference Between Two Means

Is the value of 2 1 significantly different from 0?This test gives you the answer:

If the t value is greater than 1.96, the difference between the means is significantly different from zero at an alpha of .05 (or a 95% confidence level).

The difference between the two means

the estimated standard error of the difference

21

21

21

)2(

YY

NN S

YYt

The critical value of t will be higher than 1.96 if the total N is less than 122. See Appendix C for exact critical values when N < 122.

€

SY 1 −Y 2

=(N1 −1)s1

2 + (N2 −1)s22

(N1 + N2) − 2

N1 + N2

N1N2

Estimated Standard Error of the Difference Between Two Means and Degrees of Freedom Assuming Equal Variances


€

Degrees of Freedom

df = (N1 + N 2) − 2

t-Test and Confidence Intervals


21

21

21

)2(

YY

NN S

YYt

The t-test is essentially creating a confidence interval around the difference score. Rearranging the above formula, we can calculate the confidence interval around the difference between two means:

)(2121 YY

StYY If this confidence interval overlaps with zero, then we cannot be certain that there is a difference between the means for the two samples.

Why a t Score and not a Z Score?


• Use of the Z distribution assumes the population standard error of the difference is known. In practice, we have to estimate it and so we use a t score.

• When N gets larger than 50, the t distribution converges with a Z distribution so the results would be identical regardless of whether you used a t or Z.

• In most sociological studies, you will not need to worry about the distinction between Z and t.

)(2121 YY

StYY

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