Week 9 Chapter 9 - Hypothesis Testing II: The Two-Sample Case

Week 9

Chapter 9 - Hypothesis Testing II:

The Two-Sample Case

Chapter 9

Hypothesis Testing II:

The Two-Sample Case

In This Presentation

The basic logic of the two-sample case Test of significance for sample means (large

and small samples) Tests of significance for sample proportions

(large samples) The difference between “statistical

significance” and “importance”

In The Last Chapter

Population =Penn State University

Group 1 = All Education

MajorsRS of 100

Education

Majors

In This Chapter

Population =Pennsylvania

Group 1 = All Males in

Population

Group 2 = All Females

in Population

RS of 100 Males

RS of 100 Females

Basic Logic

We begin with a difference between sample statistics (means or proportions)

The question we test: “Is the difference between the samples large

enough to allow us to conclude (with a known probability of error) that the populations represented by the samples are different?” (p. 212)

Basic Logic

The Ho is that the populations are the same There is no difference between the

parameters of the two populations If the difference between the sample statistics

is large enough, or, if a difference of this size is unlikely, assuming that the Ho is true, we will reject the Ho and conclude there is a difference between the populations

Basic Logic

The Ho is a statement of “no difference”

The 0.05 level will continue to be our indicator of a significant difference

The Five Step Model

1. Make assumptions and meet test requirements

2. State the null hypothesis, Ho

3. Select the sampling distribution and establish the critical region

4. Calculate the test statistic

5. Make a decision and interpret the results

The Five Step Model

Changes from the one-sample case: Step 1: In addition to samples selected

according to EPSEM principles, samples must be selected independently of each other: independent random sampling

Step 2: Null hypothesis statement will state that the two populations are not different

Step 3: Sampling distribution refers to difference between the sample statistics

Scenario #1: Dependent variable is measured at the interval/ratio level with large sample sizes and unknown population standard deviations

Hypothesis Test for Means

Two-Sample Test of Means (Large Samples) Do men and women significantly differ on their support

of gun control? For men (sample 1):

Mean = 6.2 Standard deviation = 1.3 Sample size = 324

For women (sample 2): Mean = 6.5 Standard deviation = 1.4 Sample size = 317

Step 1: Make Assumptions and Meet Test Requirements Independent random sampling

The samples must be independent of each other Level of measurement is interval-ratio

Support of gun control is assessed with an interval-ratio level scale, so the mean is an appropriate statistic

Sampling distribution is normal in shape Total N ≥ 100 (N1 + N2 = 324 + 317 = 641) so the

Central Limit Theorem applies and we can assume a normal shape

Step 2: State the Null Hypothesis

Ho: μ1 = μ2

The null hypothesis asserts there is no difference between the populations

H1: μ1 μ2

The research hypothesis contradicts the Ho

and asserts there is a difference between the populations

Step 3: Select the Sampling Distribution and Establish the Critical Region

Sampling Distribution = Z distribution Alpha (α) = 0.05 (two-tailed) Z (critical) = ±1.96

Step 4: Compute the Test Statistic

1. Calculate the pooled estimate of the standard error

2. Calculate the obtained Z score

Step 5: Make a Decision

The obtained test statistic (Z = -2.80) falls in the critical region so reject the null hypothesis

The difference between the sample means is so large that we can conclude (at alpha = 0.05) that a difference exists between the populations represented by the samples

The difference between men’s and women’s support of gun control is significant

Factors in Making a Decision

The size of the difference (e.g., means of 6.2 and 6.5)

The value of alpha – the higher the alpha, the more likely we are to reject the Ho

The use of one- vs. two-tailed tests (we are more likely to reject with a one-tailed test)

The size of the sample (N) – the larger the sample the more likely we are to reject the Ho

Scenario #2: Dependent variable is measured at the interval/ratio level with small sample sizes and unknown population standard deviations

Scenario #2

For small samples, s is too biased an estimator of σ so do not use standard normal distribution

Use Student’s t distribution

Hypothesis Test for Means

Two-Sample Test of Means (Small Samples) Do families that reside in the center-city have more

children than families that reside in the suburbs? For suburbs (sample 1):

Mean = 2.37 Standard deviation = 0.63 Sample size = 42

For center-city (sample 2): Mean = 2.78 Standard deviation = 0.95 Sample size = 37


The samples must be independent of each other Level of measurement is interval-ratio

Number of children can be treated as interval-ratio Population variances are equal

As long as the two samples are approximately the same size, we can make this assumption

Sampling distribution is normal in shape Because we have two small samples (N < 100),

we have to add the previous assumption in order to meet this assumption


Ho: μ1 = μ2


H1: μ1 < μ2




Sampling Distribution = t distribution Alpha (α) = 0.05 (one-tailed) Degrees of freedom = N1 + N2 -2 = 77

t (critical) = -1.671






The obtained test statistic (t = -2.16) falls in the critical region so reject the null hypothesis

The difference between the sample means is so large that we can conclude (at alpha = 0.05) that a difference exists between the populations represented by the samples

The difference between the number of children in center-city families and the suburban families is significant

Scenario #3: Dependent variable is measured at the nominal level with large sample sizes and unknown population standard deviation

Hypothesis Test for Proportions

Two-Sample Test of Proportions (Large Samples) Do Black and White senior citizens differ in their

number of memberships in clubs and organizations? (Using the proportion of each group classified as having a “high” level of membership)

For Black senior citizens (sample 1): Proportion = 0.34 Sample size = 83

For White senior citizens (sample 2): Proportion = 0.25 Sample size = 103


The samples must be independent of each other Level of measurement is nominal

We have measured the proportion of each group classified as having a “high” level of membership

Sampling distribution is normal in shape Total N ≥ 100 (N1 + N2 = 83 + 103 = 186) so the

Central Limit Theorem applies and we can assume a normal shape


Ho: Pu1 = Pu2


H1: Pu1 ≠ Pu2




Sampling Distribution = Z distribution Alpha (α) = 0.05 (two-tailed) Z (critical) = ±1.96






The obtained test statistic (Z = +1.29) does not fall in the critical region so fail to reject the null hypothesis

The difference between the sample means is small enough that we can conclude (at alpha = 0.05) that no difference exists between the populations represented by the samples

The difference between the memberships of Black and White senior citizens is not significant

Limitations of Hypothesis Testing

Significance Versus Importance As long as we work with random samples, we must

conduct a test of significance Significance is not the same thing as importance

Differences that are otherwise trivial or uninteresting may be significant

Conclusion Scenario #1: Dependent variable is measured at the

interval/ratio level with large sample sizes and unknown population standard deviations Use standard Z distribution formula

Scenario #2: Dependent variable is measured at the interval/ratio level with small sample sizes and unknown population standard deviations Use standard T distribution formula

Scenario #3: Dependent variable is measured at the nominal level with large sample sizes and known population standard deviations Use slightly modified Z distribution formula

Scenario #4: Dependent variable is measured at the nominal level with small sample sizes This is not covered in your text or in class

USING SPSS

On the top menu, click on “Analyze” Select “Compare Means” Select “Independent Samples T Test OR Paired Samples

T Test”

Test on two means (independent samples) Analyze / Compare means / Independent

samples t-test

Specify which groups you are comparing

OutputIndependent Samples Test

1.244 .265 -2.270 394 .024 -27.1146 11.94454 -50.59764 -3.63162

-2.194 251.912 .029 -27.1146 12.35795 -51.45270 -2.77657

Equal variancesassumed

Equal variancesnot assumed

Amount spentF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

The null hypothesis is rejected(as the p-value is smaller than 0.05)

Documents

Week 9 Chapter 9 - Hypothesis Testing II: The Two-Sample Case