Two Independent Groups t-Test

PSYC 2317 Mark W. Tengler, M.S.

Assignment #12 Two Independent Groups t-Test

12.1 What is measured by the estimated standard error that is used for the independent measures t statistic?

12.2 One sample has n = 15 with SS = 1660, and a second sample has n = 15 with SS = 1700. a. Find the pooled variance for the two samples. b. Compute the estimated standard error for the sample mean differences (i.e.

for the two groups). c. If the sample mean difference (i.e. M1 – M2) is 8 points, is this enough to

indicate a significant difference for a two-tailed test at the .05 level? d. If the sample mean difference (i.e. M1 – M2) is 12 points, is this enough to

indicate a significant difference for a two-tailed test at the .05 level?

12.3 Siegel (1990) found that elderly people who owned dogs were less likely to pay visits to their doctors after upsetting events than were those who did not own pets. Similarly, consider the following hypothetical data. A sample of elderly dog owners is compared to a similar group (in terms of age and health) who do not own dogs. The researcher records the number o0f visits to the doctor during the past year for each person. The data are as follows:

Control Group Dog Owners 12 8 10 5 6 9 9 4 15 6 12 14

a. Is there a significant difference in the number of doctor visits between dog owners and control subjects? Use a two-tailed test with " = .05.

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Two Independent Groups or Two Independent Samples Designs Two Independent Samples t-test

I. Assumptions for t-test A. Populations

1. two populations from which the samples are selected are normal 2. two populations must have equal variances (i.e. are the same pop)

B. two random samples (each tested once) derived by: 1. Two different samples from same population (looking to see if populational

differences due to treatment or chance) OR 2. One random sample randomly assigned to two different groups (looking to see

if differences in groups are due to treatment or chance) C. Data values

1. Sample values known (mean, standard deviation) 2. Populational values (mean, standard deviation) not known

II. Diagramming your research (shows the whole logic and process of hypothesis testing)

a. Draw a picture of your research design (see diagramming your research handout). b. There are always two explanations (i.e. hypotheses) of your research results, the

wording of which depends on whether the research question is directional (one-tailed) or non-directional (two-tailed). State them as logical opposites.

c. For statistical testing, ignore the alternative hypothesis and focus on the null hypothesis, since the null hypothesis claims that the research results happened by chance through sampling error.

d. Assuming that the null is true (i.e. that the research results occurred by chance through sampling error) allows one to do a probability calculation (i.e. all statistical tests are nothing more than calculating the probability of getting your research results by chance through sampling error).

e. Observe that there are two outcomes which may occur from the results of the probability calculation (high or low probability of getting your research results by chance, depending on the alpha (α) level).

f. Each outcome will lead to a decision about the null hypothesis, whether the null is probably true (i.e. we then accept the null to be true) or probably not true (i.e. we then reject the null as false).

III. Hypotheses

A. Two-tailed (non-directional research question) 1. Alternative hypothesis (H1): The independent variable (i.e. the treatment) does

make a difference in performance between the two groups. 2. Null hypothesis (H0): The independent variable (i.e. the treatment) does not

make a difference in performance between the two groups other than by sampling error.

B. One-tailed (directional research question) 1. Alternative hypothesis (H1): The independent variable causes one group to

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perform better or less than the other. 2. Null hypothesis (H0): The independent variable causes the one group to

perform in an opposite effect than expected or no change in performance.

IV. Determine critical regions (i.e. t value cut-off for hypothesis testing) using table A-27 A. Significance level (should be given or decided prior to experiment)

1. α or p = .05, .01, or .001 B. One- or two-tailed test

1. One-tailed: use the first row across the top 2. Two-tailed: use the second row across the top

C. Degrees of freedom 1. df = df1 + df2 = (n1 - 1) + (n2 - 1) = n1 + n2 - 2

D. With degrees of freedom and one- or two-tailed p values, find the critical t value 1. If two-tailed, then critical t value is ± t value 2. If one-tailed, then determine if critical t value is + or - t

V. Calculate t-test statistic A. t-test formula for two independent groups

t = (M1 - M2) standard error

B. Calculations 1. Compute variance (i.e. pooled variance)

s2p = [ ∑ 𝑥1

2− (∑ 𝑥1)

2

𝑛1 ] + [ ∑ 𝑥2

2− (∑ 𝑥2)

2

𝑛2 ]

(𝑛1−1)+ (𝑛2−1) =

𝑆𝑆1+ 𝑆𝑆2

𝑑𝑓1+ 𝑑𝑓2

2. Compute standard error (average distance between sample & pop means)

𝑠(𝑚1−𝑚2) = √ 𝑠𝑝

2

𝑛1 +

𝑠𝑝 2

𝑛2

3. Compute t-test statistic

t = 𝑀1− 𝑀2

𝑠(𝑚1−𝑚2)

C. Compare calculated t-statistic to critical t-value (from A-27 table) & make decision 1. Reject null and accept alternative or 2. Accept null

VI. Reporting the results of an independent measures t test

“The group that experience treatment one performed better (M = 25, SD = 4.22) than the group that experienced treatment two (M = 19, SD = 4.71). This difference was significant, t(18) = 3.00, p < .05, two-tailed.”