Chapter 6 Preview – Generalizing from Samples to Populations

Sampling error always occurs.Standard error of the mean allows

us to construct confidence intervals for means/proportions.

If the population standard deviation is known, use Z-scores.

However, this is rarely the case which is why we use T-ratios.

Chapter 7 Testing Differences between Means

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Testing Differences Between Means Establish hypothesis

about populations, collect sample data, and see how likely the sample results are, given the hypothesis. Example: Memory

enhancement N = 10

Method A Method B

82 78

83 77

82 76

80 78

83 76

Mean = 82 Mean = 77

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Method A Method B

90 70

98 90

63 91

74 56

85 78

Mean = 82 Mean = 77

Now suppose instead that the following sets of scores produced the two sample means of 82 and 77.

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The Null Hypothesis

No difference between means An obtained difference between two sample means

does not represent a true difference between their population means

Mean of the first population = mean of the second population

Example: Germans are no more obedient to authority than Americans.

Example: Female and Male teachers have the same disciplinary styles

Retain or reject the null hypothesis

Null hypothesis shown as H0: 21

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The Research Hypothesis Differences between groups, whether

expected on theoretical or empirical grounds, often provide the rationale for research

Mean of the first population does not equal the mean of the second population

If we reject the null hypothesis, we automatically accept the research hypothesis that a true population difference does exist.

Different means Example: Germans differ from Americans with

respect to obedience to authority. Research hypothesis shown as H1: 21

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Levels of Significance

To establish whether our obtained sample difference is statistically significant – the result of a real population difference and not just sampling error – it is customary to set up a level of significance

Denoted by the Greek letter alpha (α) The alpha value is the level of probability at

which the null hypothesis can be rejected with confidence and the research hypothesis can be accepted with confidence.

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Type I and II Errors

Correct Decision Type I Error P (Type I Error) = alpha

Type II Error P (Type II Error) = beta

Correct Decision

DECISION

Retain Null Reject Null

Null is true

REALITY

Null is false

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Choosing a Level of Significance Type I error and Type II error are inversely

related: the larger one error is, the smaller the other

The larger the chosen level of significance (say, .05 or even .10), the larger the chance of Type I error and the smaller the chance of Type II error.

The smaller the chosen significance level (say, .01 or even .001), the smaller the chance of Type I error, but the greater the likelihood of Type II error.

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What is the Difference Between P and Alpha? The difference between P and alpha

can be a bit confusing P is the exact probability that the null

hypothesis is true in light of the sample data

The alpha value is the threshold below which is considered so small that we decide to reject the null hypothesis

A = .001A = .01A = .05

P < .001P < .01P < .05

Standard Error of the Difference between Means

Standard deviation of the distribution of differences can be estimated.

The standard error of the differences between means is shown as:

21 XXs

21

21

21

222

211

221 NN

NN

NN

sNsNs

XX

Testing the Difference between Means

Why use t instead of z? We don’t know the true population standard deviation.

Test differences between means using t:

This is referred to as our T computed

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21

XXs

XXt

Comparing our T value

Using Table C, we find our T critical value. To calculate the degrees of freedom (df)

when testing the difference between means we use the following formula

df = N1 + N2 – 2

Alpha value is given (.05 or .01) If T computed > T critical, reject null If T computed < T critical, accept null

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Testing the Difference between Means Suppose that we

obtained the following data for a sample of 25 liberals and 35 conservatives on the permissiveness scale.

Calculate the estimate of the standard error of the differences between means.

Then, translate the difference between sample means into a t ratio.

Liberals Conservatives

N1 = 25 N2 = 35

Mean1 = 60 Mean2 = 49

S1 = 12 S2 = 14

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Continued. If necessary, find the mean and standard deviation first. Otherwise:

Step 1: Find the standard error of the difference between means.

Step 2: Compute the t ratio. Step 3: Determine the critical value for t. Step 4: Compare the calculated and table t values.

End Day 1

http://www.people-press.org/files/legacy-pdf/10-15-12%20Debates%20Release%20.pdf

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Comparing the Same Sample Measured Twice So far, we have discussed making

comparisons between two independently drawn samples

Before-after or panel design: the case of a single sample measured at two different points in time (time 1 vs. time 2)

For example, a polling organization might interview the same 1,000 Americans both in 1995 and 2000 in order to measure their change in attitude over time.

Numerous uses for this type of test

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Test of Difference between Means for Same Sample Measured Twice

Suppose that several individuals have been forced by a city government to relocate their homes to make way for highway construction.

As researchers, we are interested in determining the impact of forced residential mobility on feelings of neighborliness.

What would the null and research hypotheses state?

We interview a random sample of 6 individuals about their neighbors both before and after they are forced to move.

Testing the Difference Between Means for the Same Sample Measured Twice To obtain the standard error of the

difference between means use the following formula:

Where: SD = Standard deviation of the distribution of

before-after difference scores.

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Finding the t ratio

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Computed T ratio: Critical T:df = N – 1α = .05 or .01

Use Table C

Compare the computed T with the critical T.If |T| > critical T, reject null hypothesis.If |T| < critical T, retain null hypothesis.

Their Scores

Respondent Before After

Stephanie 2 1

Leon 1 2

Carol 3 1

Jake 3 1

Julie 1 2

David 4 1

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Test of Difference between Means for Same Sample Measured TwiceMake sure to first state the null and research hypothesis! After finishing the table: Step 1: Find the mean for each point in time

(Before = X1; After = X2) Step 2: Find the standard deviation for the

difference between time 1 and time 2 Step 3: Find the standard error of the difference

between means Step 4: Calculate the t score Step 5: Find the critical t score Step 6: Compare the obtained t ratio with the

critical t score

Two Sample Test of Proportions As in Chapter 6, we are interested in testing the

difference between two groups measured in proportions. Males/Females, Blacks/Whites,

Liberals/Conservatives, Violent/Nonviolent criminals, Adult/Juvenile offenders, etc

Use Z scores for critical values When alpha = .05, Z score of 1.96 is used When alpha = .01, Z score of 2.58 is used

Use for stating the null/research hypotheses

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Two Sample Test of Proportions Formulas

Step 1: Find P* (combined sample proportion).

P* Step 2: Standard error of the difference of proportions.

Sp-p =

Step 3: Find the Z computed score.

z =

Step 4: Compare Z computed with Z critical & interpret.

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Two Sample Test Example

A criminal justice researcher is interested in marijuana usage and driving while high of upper level undergraduates in her particular school. After taking a random sample of 300 students, she discards any surveys of students who have not smoked marijuana. She is left with the following data:

Test the research hypothesis at the alpha level of .05. What do your results indicate?

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Male Female

Sample Size 127 149

Driven high 56 36

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One-Tailed Tests

1. A one-tailed test rejects the null hypothesis at only one tail of the sampling distribution.

2. It should be emphasized, however, that the only changes are in the way the hypotheses are stated and the place where the t table is entered.

3. Used when the researcher anticipates the direction of change.

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One-Tailed Tests Cont.1. A one-tailed test is appropriate when a researcher is

only concerned with a chance (for a sample tested twice) or difference (between two independent samples) in one pre-specified direction or when a researcher anticipates the DIRECTION of the change or difference.

1. Example: An attempt to show that Black defendants receive harsher sentences (mean sentence) than Whites indicates the need for a one-tailed test.

1. Null: u1 ≥ u2 White defendants receive equal or harsher sentences.

2. Research: u1 < u2 Black defendants receive harsher sentences

2. If we were only attempting to show that there are differences in sentencing by race indicates the need for a two tailed test.

2. If however, a researcher is just looking for differences in sentencing by race, whether it is Blacks or Whites who get harsher sentences, he or she would instead use a two-tailed test.

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Requirments for Testing the Differences between Means

1. A comparison between two means

2. Interval data

3. Random sampling

4. A normal distribution

5. Equal population variances

Summary

Testing hypotheses about differences between sample means

Null / Research hypothesis Type I and Type II errors Alpha levels used to reject or retain the null

hypothesis