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11/7/2017

Chapter 9: Testing the Difference Between Two

Means

• In Chapter 7, we compared to μ, where one sample was drawn from one population

• In this chapter, we’ll test the difference between two samples drawn from two populations

Testing the Difference Between Two Means

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

An Example…

• Will people take longer to leave a parking space when another driver is waiting (“Intruder”) compared to when no other driver is waiting (“No intruder”)?

• Observational study with 15 drivers in each condition (Ruback & Juieng, 1997)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

An Example…

Intruder No Intruder

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

• Bar chart − One nominal variable as IV − Bar height represents mean on DV − Error bars represent 1 standard error of the mean

above and below the mean (± 1 s )

An Example…

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

An Example…

• Are the two groups different from each other?

– Chapter 7: evaluated a sample mean using a distribution of sample means (the sampling distribution of the mean)

– Now: Evaluate the difference between two sample means using a distribution of differences between sample means (the sampling distribution of the difference).

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Sampling Distribution of the Difference

• Sampling distribution of the difference: distribution of all possible differences between two sample means when an infinite number of pairs of samples of size N are randomly drawn from two populations

– Used to determine the probability of obtaining any particular difference between two sample means

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Sampling Distribution of the Difference

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Lets assume that two populations are not different

• We randomly draw samples from each population and calculate differences between the two sample means

• Although we expect there to be no (zero) differences between the sample means, because of sampling error, we will get a distribution of differences

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The Sampling Distribution of the Difference: Characteristics

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Modality − Mean = 0

• Symmetry − Approximately normal − Shape determined by sample size

• Variability − Standard error of the difference

Inferential Statistics: Testing the Difference between Two Sample Means

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means

• State the null and alternative hypotheses

H0: µIntruder = µNo intruder H1: µIntruder  µNo intruder

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample Means

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means

• Make a decision about the null hypothesis

– Calculate the degrees of freedom (df)

– Set alpha (α), identify the critical values, and state a decision rule

– Calculate a statistic: t‐test for independent means

– Make a decision whether to reject the null hypothesis

– Determine the level of significance

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample Means

• Make a decision about the null hypothesis

– Calculate the degrees of freedom (df)

– Because we have two samples from two populations,

df = (N1 – 1) + (N2 – 1)

= (15 – 1) + (15 – 1)

= 28

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means

• Make a decision about the null hypothesis

– Set alpha (α), identify the critical values, and state a decision rule

• For α = .05 (two‐tailed) and df = 28, critical value =  2.048

• If t < ‐2.048 or > 2.048, reject H0; otherwise, do not reject H0

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Make a decision about the null hypothesis.

− Calculate a statistic: t‐test for independent means

Inferential Statistics: Testing the Difference between Two Sample Means

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Make a decision about the null hypothesis − Calculate the standard error of the

difference

Inferential Statistics: Testing the Difference between Two Sample Means

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Make a decision about the null hypothesis. − Calculate the t‐statistic for independent

means:

t = 2.42

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Inferential Statistics: Testing the Difference between Two Sample Means

• Make a decision about the null hypothesis

– Make a decision whether to reject the null hypothesis

t = 2.42 > 2.048  reject H0 (p < .05) – Determine the level of significance

• For  = .01, critical value = 2.763 • t = 2.42 < 2.763  p < .05 (but not < .01)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample Means

• Make a decision about the null hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample Means

• Draw a conclusion from the analysis – The mean departure time for the 15 drivers in the Intruder group (M = 40.73s) is significantly greater than the mean departure time for the 15 drivers in the No intruder group (M = 31.67s), t(28) = 2.42, p < .05.

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means

State the null and alternative hypotheses (H0 and H1)

Make a decision about the null hypothesis

Draw a conclusion from the analysis

Relate the result of the analysis to the research hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample Means

• Relate the result to the research hypothesis

– “The present series of studies is consistent with prior findings that people display territorial defense in public territories...What is new about the present research is that it suggests people sometimes display territorial behavior merely to keep others from possessing the space even when it no longer has any value to them” (Ruback & Juieng, 1997, p. 831).

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means

• Assumptions of the t‐test for independent means

– Assumption of normality: the distribution of scores in the two populations from which the samples are drawn are normal

– Homogeneity of variance: variance of scores in the two populations is the same

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample Means

• If the assumptions are violated, researchers may make the wrong decision regarding the null hypothesis

• However, the t‐statistic is robust – Able to withstand moderate violations of the assumptions

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

Inferential Statistics: Testing the Difference between Two Sample Means (unequal sample sizes)

• Example: − Researchers tested the effects of a healthy living and exercise intervention on kindergarten and first grade students’ (N1 = 16) ability to jump rope for 30 seconds. They compared these scores to a control group (N2 = 11) who did not receive the intervention.

− Data from Matvienko and Ahrabi‐Fard (2010)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Intervention (N1 = 16) Control (N2 = 11)

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

11/7/2017

• State the null and alternative hypotheses

H0: µIntervention= µControl H1: µIntervention  µControl

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

• Make a decision about the null hypothesis

– Calculate the degrees of freedom (df)

df = (N1 – 1) + (N2 – 1)

= (16 – 1) + (11 – 1)

=25

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

11/7/2017

• Make a decision about the null hypothesis – Set alpha (α), identify the critical values, and state a decision rule

• For α = .05 (two‐tailed) and df = 25, critical value =  2.060

• If t < ‐2.060 or > 2.060, reject H0; otherwise, do not reject H0

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

• Make a decision about the null hypothesis − Calculate a statistic: t‐test for independent

means:

11/7/2017

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

• Make a decision about the null hypothesis − Calculate the standard error of the difference

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

11/7/2017

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

• Make a decision about the null hypothesis − Calculate a statistic: t‐test for independent

means:

t = 3.97

• Make a decision about the null hypothesis

– Make a decision whether to reject the null hypothesis

t = 3.97 > 2.060  reject H0 (p < .05) – Determine the level of significance

• For  = .01, critical value = 2.787 • t = 3.97 > 2.787  p < .01

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

11/7/2017

• Make a decision about the null hypothesis

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

• Draw a conclusion from the analysis – The average number of rope jumps in 30 seconds is significantly greater for the 16 students who received the intervention (M = 27.31) than for the 11 students in the Control group who did not receive the intervention (M = 11.91), t(25) = 3.97, p < .01.

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

11/7/2017

• Relate the result to the research hypothesis

– “This finding suggests that programs emphasizing the enhancement of basic motor skills that children apply in a variety of games and sports may be an effective approach to increasing overall activity and fitness levels of young children” (Matvienko & Ahrabi‐Fard, 2010, p. 303).

Inferential Statistics: Testing the Difference between Two Sample

Means (unequal N)

Testing the Difference Between Two Sample Means

• In the examples we’ve discussed so far, we’ve compared samples from two different populations

• These would be considered between‐ subjects research designs

– Each participant appears in only one group

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Inferential Statistics: Testing the Difference between Paired Means

• Within‐subjects research designs test differences (or change) within the same participant

– Differences within a person regarding different situations

– Repeated administrations (longitudinal designs)

– Pre‐test – post‐test design

Inferential Statistics: Testing the Difference between Paired Means

• Example: − A sample of 20 parents were tested on their knowledge of childhood depression and anxiety. Three weeks later, following a web‐based program, the same parents were re‐tested on their knowledge

− Data from Deitz et al. (2009)

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Inferential Statistics: Testing the Difference between Paired Means

Pre‐test Post‐test

Inferential Statistics: Testing the Difference between Paired Means

• Consequences of the same people appearing in both conditions

– We explicitly identify the paired data by calculating a difference between the two scores

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Inferential Statistics: Testing the Difference between Paired Means

• State the null and the alternative hypotheses

H0: µD = 0

H1: µD  0 µD: Mean difference between the two scores

µD = 0 is the same as µPre‐test = µPost‐test

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Inferential Statistics: Testing the Difference between Paired Means

• Make a decision about the null hypothesis

– Calculate the degrees of freedom (df)

df = (ND – 1)

= (20 – 1)

= 19

Inferential Statistics: Testing the Difference between Paired Means

• Make a decision about the null hypothesis

– Set alpha (α), identify the critical values, and state a decision rule

• For α = .05 (two‐tailed) and df = 19, critical value =  2.093

• If t < ‐2.093 or > 2.093, reject H0; otherwise, do not reject H0

11/7/2017

Inferential Statistics: Testing the Difference between Paired Means

• Make a decision about the null hypothesis − Calculate a statistic: t‐test for dependent

means:

Inferential Statistics: Testing the Difference between Paired Means

• Make a decision about the null hypothesis − Calculate standard error of the difference

scores:

11/7/2017

Inferential Statistics: Testing the Difference between Paired Means

• Make a decision about the null hypothesis − Calculate t‐statistic for dependent means

Inferential Statistics: Testing the Difference between Paired Means

• Make a decision about the null hypothesis

– Make a decision whether to reject the null hypothesis

t = ‐5.77 < ‐2.093 reject H0 (p < .05) – Determine the level of significance

• For  = .01, critical value for t = ‐2.861 • t = ‐5.77 < ‐2.861 p < .01

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Inferential Statistics: Testing the Difference between Paired Means

• Make a decision about the null hypothesis

Inferential Statistics: Testing the Difference between Paired Means

• Draw a conclusion from the analysis: – The average knowledge scores for the 20 parents were significantly higher after completing the web‐based intervention program (M = 21.15) than before beginning the program (M = 15.55), t(19) = ‐5.77, p < .01.

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Inferential Statistics: Testing the Difference between Paired Means • Relate the result to the research hypothesis – “These findings indicate that the program can be an effective intervention for improving parents’ knowledge of children’s mental health problems and boost their confidence in handling such issues …The study findings lend support to the growing literature on the utility of offering web‐ based programs to improve the health of the general population” (Dietz et al, 2009, p. 492).

Inferential Statistics: Testing the Difference between Paired Means

• Assumptions of the t‐test for dependent means

– Assumption of normality • Larger sample sizes make meeting this assumption more likely

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Looking Ahead • In this chapter, we explored the process of hypothesis testing using a situation slightly more complicated than those presented in earlier chapters. However, we still use the same basic steps in hypothesis testing.

• Because we must rely on probability, there exists the possibility that the decision made about the null hypothesis may be in error. The next chapter discusses these errors in greater detail, as well as what researchers can do to minimize the possibility and impact of making these errors.