Chapter 10: Independent-Samples t Test

Question 1

Which of the following is appropriate for an Independent-Samples t-test:

A: Within-group research design

B: Between-group research design

Answer 1

B: Between-group research design

> And don’t forget that an independent-samples t-test is only for two groups

Question 2

Define a between-group research design:

A: A between-group research design involves studying a single group of participants over an extended period, observing how their behavior changes within different conditions. This is distinct from within-group designs, where multiple groups are examined simultaneously.

B: A between-group research design seeks to compare two or more groups of participants. Unlike the within-group design, each condition is comprised of different participants.

C: In a between-group research design, researchers focus on the changes in participants’ behaviors within the same group across various conditions. This contrasts with within-group designs, where distinct groups are established for each experimental condition.

Answer 2

B: A between-group research design seeks to compare two or more groups of participants. Unlike the within-group design, each condition is comprised of different participants.

> Imagine you have a sample of participants, with a between-group research design participants are randomly split into different groups, there’s no crossover.

Examples:
> In an experimental application participants are randomly assigned to either a treatment or a control condition (treatment vs. control).
> In a non-experimental application participants are divided into groups based on some pre-existing characteristic (old vs. young).

Within vs. between:
> Within typically requires fewer participants and you can experience carry-over which is a con
> Between typically requires more participants and randomization is critical

Question 3

Define an independent-samples t-test:

A: The independent-samples t-test is designed for within-group experiments, comparing the means of the same group before and after exposure to different conditions. It is particularly effective when analyzing the variation in scores within a single group.

B: An independent-samples t-test is a statistical method used to compare the means of two related groups in a within-group design. It is employed when conditions are applied to the same group of participants, and researchers are interested in understanding the variation in scores within these conditions.

C: The independent-samples t-test is appropriate for between-group designs with two groups. The statistic of interest is the mean difference between the two groups.

Answer 3

C: The independent-samples t-test is appropriate for between-group designs with two groups. The statistic of interest is the mean difference between the two groups.

NOTE: The research question itself cannot tell you whether it’s referring to a within or between-group design. Only the details of the study can tell you this.

Question 4

So, what’s the different between the 3 types of t-tests?

PUT THIS ON YOUR CHEAT SHEET!!!

Answer 4

> One-sample t-test: Test whether the known or hypothesized value of the mean is true and supported by the sample mean. Basically, you’re comparing one group to the population. SAT scores at UCLA (sample) compared to SAT scores in the USA (population).

Paired-sample t-test: Test the difference between two
conditions and each individual contributes two scores (within-group design only). For example, testing exam scores in 100A before and after using a cheat sheet (pre vs. post).

Independent-samples t-test: Test the difference between two conditions and each individual contributes one score. For example, exam scores for college graduates vs. exam scores for non-college graduates.

NOTE: Both the paired-sample t-test and independent-
samples t-test can answer the same research question.
It’s the research design that determines which test to
use. You’re looking for clues like, is it a within-group design where you’re testing pre vs. post (paired-sample), or a between-group design where you’re testing participants in separate groups (independent-sample).

Question 5

Han wants to know whether there is a difference in the average sleep duration between students majoring in Psychology and those majoring in Sociology. Which t-test should she use?

A. One-sample t-test
B. Paired-sample t-test
C. Independent-sample t-test

Answer 5

C. Independent-sample t-test

> This is clearly a between-group research design where there are two separate groups, no crossover, and each individual contributes one score.

Question 6

Han wants to know whether students pursuing a major in Psychology experience an average of 7 hours of sleep. Which t-test should she use?

A. One-sample t-test
B. Paired-sample t-test
C. Independent-sample t-test

Answer 6

A. One-sample t-test

> You’re comparing Psychology students (sample) to a larger population and each individual will contribute one score.

Question 7

Han wants to know whether there is a change in students’ sleep hours before and after the midterm exams. Which t-test should she use?

A. One-sample t-test
B. Paired-sample t-test
C. Independent-sample t-test

Answer 7

B. Paired-sample t-test

> This is clearly a within-group research design where each individual is tested pre-midterm and then again post-midterm and provides two scores.

Question 8

What is the null hypothesis for an independent-sample t-test?

A: The null hypothesis for independent-sample t-test between-group designs targets the mean difference between groups. A typical application specifies a null hypothesis of no difference (nothing happening).

B: The null hypothesis for an independent-sample t-test asserts that the means of the two groups are exactly the same. It presupposes that there is a perfect match in the performance of participants across different conditions.

C: In an independent-sample t-test, the null hypothesis states that there is a significant difference between the means of the two groups. Researchers use this to test the idea that any observed difference is not due to chance but reflects a true disparity in the population.

Answer 8

A: The null hypothesis for independent-sample t-test between-group designs targets the mean difference between groups. A typical application specifies a null hypothesis of no difference (nothing happening).

H0: µ1 - µ2 = 0

Each group is equal, there’s no difference between the two!

SHE MIGHT ALSO SAY:
A hypothesis about the population
mean that opposes the researcher’s belief, usually
specifying the idea that “nothing is going on”.

Question 9

What is the alternate hypothesis for a one-tailed independent-sample t-test?

A: In a one-tailed independent-sample t-test, the alternative hypothesis suggests that there is no difference between the two groups.

B: The alternative hypothesis for a one-tailed independent-sample t-test states that the means of the two groups are equal.

C: A one-tailed hypothesis specifies that one group is
higher than the other.

Answer 9

C: A one-tailed hypothesis specifies that one group is
higher than the other.

Ha: µ1 > µ2 OR Ha: µ1 < µ2

SHE MIGHT ALSO SAY:
The hypothesis about the population mean that reflects the researcher’s interests or beliefs.

Question 10

What is the alternate hypothesis for a two-tailed independent-sample t-test?

A: In a two-tailed independent-sample t-test, the alternative hypothesis asserts that the means of the two groups are exactly the same.

B: A typical application specifies a two-tailed hypothesis
where the groups differ (e.g., group one and group two are not the same).

C: The alternative hypothesis for a two-tailed independent-sample t-test states that the two groups are similar in terms of their means. This means that researchers are interested in showing that there is no significant difference between the groups, whether in the positive or negative direction.

Answer 10

B: A typical application specifies a two-tailed hypothesis
where the groups differ (e.g., group one and group two are not the same).

Ha: µ1 - µ2 ≠ 0

SHE MIGHT ALSO SAY:
The hypothesis about the population mean that reflects the researcher’s interests or beliefs.

Question 11

What is the standard error of an independent-sample t-test?

A: The standard error in an independent-sample t-test is the average variability of scores within each group. It represents the spread of individual data points within the groups, providing an overall measure of dispersion.

B: The standard error is the standard deviation of
the mean difference from many different two-group samples.

C: In an independent-sample t-test, the standard error is the standard deviation of the difference between the means of the two groups. It quantifies the extent to which the means of the groups deviate from each other.

Answer 11

B: The standard error is the standard deviation of
the mean difference from many different two-group samples.

A.K.A:
> Average standard error of the mean difference
> Pooled standard deviation
> Sampling distribution of the group average mean difference

> You should include the formula image from slides 27 and 28 on your cheatsheet

> Sample size must be the same when comparing one study to another (for instance Ben and Han each conducted a study with the same sample size - 100 participants each) BUT group sizes do not need to be the same within an individual study (for instance Ben has 50 men and 50 women = 100, and Han has 45 men and 55 women = 100).

Question 12

How do you interpret standard error for an independent-sample t-test?

ADD THIS TO YOUR CHEATSHEET!

Answer 12

Example: The standard deviation of the mean differences from many different random samples (the sampling distribution) is .140.

OR….

Example: The average/expected difference between the sample mean difference and the true population mean difference is .140

OR…

Example: On average, two random samples of sizes of 99 and 92 should have a sample mean difference that differs from zero (the hypothesized population mean difference) by 0.140

Question 13

What is a t-statistic for an independent-sample t-test?

A: A t-statistic converts the difference between the
estimate and hypothesis to a standardized metric

B: The t-statistic in an independent-sample t-test is the actual difference between the means of the two groups. It is a raw measure of the disparity between the sample estimates and the hypothesized population values.

C: In an independent-sample t-test, the t-statistic is the probability of observing the obtained difference between the groups by chance. It quantifies the likelihood that the observed mean difference occurred randomly in the sample.

Answer 13

A: A t-statistic converts the difference between the
estimate and hypothesis to a standardized metric

> Represented as standard error units

> Represents the standard error of difference

> YOU NEED TO ADD THE FORMULA IMAGE ON SLIDES 34 AND 35 TO YOUR CHEATSHEET

SHE MIGHT ALSO SAY:
The difference between the sample mean and the
hypothesized population mean in standard error units. Exactly like a z score but applied to an estimate rather than an individual’s score.

Question 14

How do you interpret a t-statistic for an independent-sample t-test?

ADD THIS TO YOUR CHEATSHEET!

Answer 14

Example: The sample mean difference is 1.22 standard error units higher than the hypothesized population mean difference of 0

OR….

Example: The sample group mean difference is 1.22 times as large as what we would expect due to sampling error alone

Question 15

What are the 3 equivalent ways to test the null hypothesis?

A: t-statistic, p-value, and confidence intervals

B: t-statistic, p-value, and standard deviation

C: Standard deviation, sample mean, t-statistic

Answer 15

A: t-statistic, p-value, and confidence intervals

Question 16

How do you calculate degrees of freedom in an independent-sample t-test?

A: N-1

B: N-2

C: n1 + n2 - 2

Answer 16

C: n1 + n2 - 2

It’s slightly different than we’ve done it before (N-1) because now we have more than one group we’re comparing.

> Remember that you’ll use the degrees of freedom to calculate the critical value!

Question 17

How do you interpret a critical value for an independent-sample t-test?

ADD THIS TO YOUR CHEATSHEET!

Answer 17

Example: If the null hypothesis is true, 95% of all samples we could work with should have t-statistics between about ± 1.973

Question 18

You’ll need to look at the example in the slides to answer this - Slide 43

Based on t = 1.22, what can you conclude?

A. Reject the null hypothesis
B. Fail to reject the null hypothesis

Answer 18

B. Fail to reject the null hypothesis

Question 19

You’ll need to look at the example in the slides to answer this - Slide 45

Based on the t-statistic, what can you conclude
about the p-values?

A. p > 0.05
B. P < 0.05

Answer 19

A. p > 0.05

Question 20

How do you interpret the probability (p-value) for a two-tailed independent-sample t-test?

ADD THIS TO YOUR CHEATSHEET!

Answer 20

Example: If the true difference in the population is zero, the probability of drawing a sample with a t-statistic of ± 1.22 or more extreme is .225 (22.5%)

> We always start the interpretation with… “If the true difference in the population is…”

> In a two-tailed hypothesis “more extreme” refers to both sides of the distribution

SHE MIGHT ALSO SAY:
Assuming that the null hypothesis is true, how likely is
such a population to have produced an estimate at least as different as the one from the data.

Question 21

You’ll need to look at the example in the slides to answer this - Slide 50

What is your conclusion about the statistical significance and decision about the null hypothesis?

A. Reject the null hypothesis
B. Accept the null hypothesis
C. Fail to reject the null hypothesis

Answer 21

C. Fail to reject the null hypothesis

Question 22

What is a 95% confidence interval for an independent-sample t-test?

A: It provides a measure of the variability of the data within the groups.

B: It reflects the precision of the sample mean estimate

C: The 95% confidence interval is computed as the estimate plus or minus the critical value times the
standard error

Answer 22

C: The 95% confidence interval is computed as the estimate plus or minus the critical value times the
standard error

95% C.I. = estimate ± (C.V. × standard error)

> Remember that C.V. × standard error is the margin of error

> Estimate = sample mean difference

> Used to find the population mean difference

Question 23

You’ll need to look at the example in the slides to answer this - Slide 53

Based on the confidence interval, what can you conclude about the null hypothesis (true group mean difference = zero)?

A. Reject the null hypothesis
B. Fail to reject the null hypothesis

Answer 23

B. Fail to reject the null hypothesis

Question 24

How do you interpret a 95% confidence interval for an independent-sample t-test?

ADD THIS TO YOUR CHEATSHEET!

Answer 24

We are 95% confident that the true population difference falls somewhere between −.106 (self-focused
higher by .106) and .446 (other-focused group higher
by .446)

A population difference of zero is in this range

The probability of the sample data must therefore be
p > .05 (not “significant”)

We fail to reject the null hypothesis

Question 25

So, what’s the difference between the 3 types of confidence intervals?

PUT THIS ON YOUR CHEAT SHEET!!!

Answer 25

> The 95% confidence interval in the one-sample t-test tells us the possible range of the population mean.

> The 95% confidence interval in the paired-samples t-test tells us the possible range of the population mean
difference between two conditions or the population
mean of the change scores.

> The 95% confidence interval in the independent-samples t-test tells us the possible range of the population mean difference between two conditions/groups.

Question 26

How do you interpret the probability (p-value) for a one-tailed independent-sample t-test predicting a
higher mean for the other-focused message
group.

ADD THIS TO YOUR CHEATSHEET!

Answer 26

That is Ha: µ1 > µ2

If the true difference in the population is zero, the probability of drawing a sample with a t-statistic of 1.22 or more extreme is .113 (11.3%)

And you would find this value by dividing the two-tailed p-value by 2 (unless the t-statistic is negative, then you would divide the two-tailed p-value by 2 and then subtract that number from 1

Example:
Two-tailed p-value = 0.225
0.225 ÷ 2 = > p-value = 0.1125 (if t-statistic is positive)
1-0.1125 = > p-value = 0.8875 (if the statistic is negative)

Remember that the left and right-hand side p-values must equal 1
0.1125 + 0.8875 = 1

It’s also helpful to draw a picture so you can visualize this better

** REMEMBER that switching from a two-tailed hypothesis to a one-tailed hypothesis will change your confidence interval and p-value but NOT your t-statistic and null hypothesis **

Question 27

What is the Cohen’s d Effect Size (Standardized Mean Difference) for an Independent-sample t-test?

A: Dividing the mean difference by the standard deviation expresses the group difference on the
z score metric

B: Cohen’s d Effect Size is calculated by multiplying the mean difference by the standard deviation, providing a measure of how much the groups differ in absolute terms.

C: To compute Cohen’s d Effect Size, you subtract the standard deviation from the mean difference, offering a standardized metric for the effect size in an independent-sample t-test.

Answer 27

A: Dividing the mean difference by the standard deviation expresses the group difference on the
z score metric

Mean difference ÷ average standard deviation

Represented by standard deviation units
** Always use the denominator to determine this if you forget **

It might be helpful to add the formula images from slides 61 and 62 to your cheat sheet

Question 28

MAKE SURE YOU ADD THE EFFECT SIZE GUIDELINES FROM SLIDE 63 TO YOUR CHEAT SHEET!!!

Answer 28

MAKE SURE YOU ADD THE EFFECT SIZE GUIDELINES FROM SLIDE 63 TO YOUR CHEAT SHEET!!!

Question 29

How do you interpret Cohen’s d Effect Size (Standardized Mean Difference) for an Independent-sample t-test?

ADD THIS TO YOUR CHEATSHEET!

Answer 29

The difference between the two groups is about .17 points

On a standardized metric, Cohen’s d equates this difference to .176 standard deviation units

The mean difference is negligible in magnitude

Question 30

Cohen’s d tells us ______

A. Statistical significance
B. Practical significance
C. Cohen receives a D grade

Answer 30

B. Practical significance

All effect sizes are of practical significance

Question 31

You will need to know how to interpret an APA-Style Summary so add this to your cheat sheet!!!!

Answer 31

Step 1 - Claim research style and type of study:
We used an independent-sample t-test to examine the
difference between the self-focused and other-focused
message groups.

Step 2 - Descriptive statistics:
Table 1 gives the descriptive statistics.

Step 3 - Significance:
The analysis revealed a non-significant difference

Step 4 - Stats:
t(189) = 1.22, p = .225, and the 95% confidence interval for the mean difference ranged from −.106 (self-focused higher by .106) and .446 (other-focused group higher by .446).

Step 5 - Practical significance:
Additionally, the standardized mean difference indicated that the magnitude of the effect size is consistent with Cohen’s negligible effect size benchmark (d = .176).

Question 32

I WOULD PROBABLY ALSO ADD THE APA-STYLE TABLE FROM SLIDE 67

Answer 32

I WOULD PROBABLY ALSO ADD THE APA-STYLE TABLE FROM SLIDE 67

Question 33

MAKE SURE YOU KEEP YOUR OLD CHEAT SHEET NOTES ABOUT ORDINAL, NOMINAL, RATIO, INTERVAL, CATEGORICAL, AND NUMERICAL DATA!

Answer 33

MAKE SURE YOU KEEP YOUR OLD CHEAT SHEET NOTES ABOUT ORDINAL, NOMINAL, RATIO, INTERVAL, CATEGORICAL, AND NUMERICAL DATA!

Question 34

In Jamovi - For an independent-sample t-test:

How to calculate:
> T-statistic
> P-value
> Degrees of freedom
> The individual group means, medians, standard deviations, and standard errors
> Mean difference
> Standard error difference
> Cohen’s d effect size
> Confidence interval

PUT THIS ON YOUR CHEAT SHEET!!!

Answer 34

Step 1:
> Analyses, T-Tests, Independent Samples T-Test

Step 2:
> The “message” variable tells you what group the participants were in (grouping variable) and the “other” variable is what’s being measured (the dependent variable)
> Move the “other” variable over to the “Dependent Variables” and move the “message” variable over to the “Grouping Variable”

Step 3:
> Under “Hypothesis” choose either <, >, or ≠

Step 4:
> Under “Additional Statistics” click “Mean Difference,” “Confidence Interval,” Effect Size,” and “Descriptives”

Question 35

In Jamovi - For an independent-sample t-test:

How to calculate critical value

PUT THIS ON YOUR CHEAT SHEET!!!

Answer 35

Step 1:
> Write down the degrees of freedom from the step before this OR you can do the math yourself. It’s just:
N1 + N2 -2

Step 2:
> Choose “distrACTION” and “T-Distribution”

Step 3:
> Enter the degrees of freedom under “Parameters” and df =

Step 4:
> Click “Compute Quantile(s)” and “Central Interval Quantiles

Step 5:
> Change p to “p = 0.95” and hit ENTER

The values that generate as “Quantile(s) x1 and x2 are what we’re looking for

Question 36

Assignment 5 / Question 3:

The “argMoral” variable is the perceived moral argument score based on a question: “To what extent would you say this is a moral argument?” 1 indicates “not at all” and 5 indicates “extremely”. A larger value indicates that an individual perceives the message more as a moral argument.

3. What is the measurement scale of the perceived moral argument score “argMoral”?

a) Nominal
b) Quasi-interval
c) Numeric

Answer 36

b) Quasi-interval

Even though she doesn’t say Likert scale you should know this is a Likert scale because it says,”1 indicates “not at all” and 5 indicates “extremely”.

Question 37

Assignment 5 / Question 9:

The t-statistic is 2.62, but it doesn’t really matter. This is just a question about interpreting the result of a t-statistic score and will be the same no matter what the t-statistic is.

Suppose the t statistic value in the 8th question is x. Which of the following interpretations is correct for the t statistic?

a) The other-focused group average was about x times larger than the self- focused group average

b) The self-focused group average was about x times smaller than the other- focused group average

c) On the moral arguments scale (i.e., argMoral), the other-focused and self- focused group averages differed by x points

d) The difference between the other-focused and self-focused group averages was about x times as large as the difference you would expect due to random sampling error

e) The difference between the other-focused and self-focused group averages was equivalent to x standard deviation units

Answer 37

d) The difference between the other-focused and self-focused group averages was about x times as large as the difference you would expect due to random sampling error

Question 38

Assignment 5 / Question 10:

Probability value interpretation!

The probability value (p-value) for the t-test is 0.010. Which of the following interpretations is correct?

a) The probability that the null hypothesis is true is 0.010

b) The probability of drawing a sample that has a t statistic from this sample or more extreme is 0.010

c) The probability of drawing a sample that shows a mean difference equal to the one seen in this dataset is 0.010

d) If there is truly no difference between the other-focused and self-focused groups in the population, the probability of selecting a sample that shows a t
statistic from this sample or more extreme is 0.010

e) If there is truly a difference between the other-focused and self-focused groups in the population, the probability of selecting a sample that shows a t statistic from this sample or more extreme is 0.010

Answer 38

d) If there is truly no difference between the other-focused and self-focused groups in the population, the probability of selecting a sample that shows a t
statistic from this sample or more extreme is 0.010

Question 39

Assignment 5 / Question 11:

Based on a probability value of 0.010, which of the following is true regarding the null hypothesis?

a) The estimate of the mean difference from the sample is consistent with the null hypothesis

b) The estimate of the mean difference from the sample refutes the null hypothesis

Answer 39

b) The estimate of the mean difference from the sample refutes the null hypothesis

Question 40

Assignment 5 / Question 12:

The confidence interval is 0.142 - 1.02 and I believe we’re comparing it to the null hypothesis of zero!

Obtain the confidence interval from Jamovi. Which of the following interpretations is correct for the confidence interval?

a) The confidence interval covers the population mean difference hypothesized in the null hypothesis

b) The confidence interval does not cover the population mean difference hypothesized in the null hypothesis

Answer 40

b) The confidence interval does not cover the population mean difference hypothesized in the null hypothesis

Question 41

Assignment 5 / Question 14:

Cohen’s d is 0.500 but it doesn’t really matter. This is simply a question asking you to interpret Cohen’s d and will be the same no matter what the value is.

Let’s assume the Cohen’s d statistic in Question 13 is x. Which of the following interpretations is correct for this Cohen’s d statistic?

a) On the original metric (i.e., the metric of the data, a 1-5 scale), the other- focused and self-focused group means differed by about x points

b) On a standardized metric, the other-focused and self-focused group means differed by about x standard deviation units

c) On a standardized metric, the other-focused and self-focused group means differed by about x standard error units

Answer 41

b) On a standardized metric, the other-focused and self-focused group means differed by about x standard deviation units

Question 42

Assignment 5 / Question 15:

For this question, Cohen’s d does matter and it’s 0.500

Considering the value of the standardized mean difference, which of the following interpretations is correct?

a) The group difference (mean difference) is negligible

b) The group difference (mean difference) is consistent with a small effect

c) The group difference (mean difference) is consistent with a moderate effect

d) The group difference (mean difference) is consistent with a large effect

Answer 42

c) The group difference (mean difference) is consistent with a moderate effect

Question 43

Assignment 5 / Question 17:

P-value is 0.005 and it was for a > (greater than) one-tailed hypothesis

Suppose the p-value in the 16th question is x. How should we interpret the p- value in this one-tailed test?

a) If there is truly no difference in the sample, the probability of selecting a sample that shows a t statistic from this sample or more extreme is x

b) If there is truly no difference in the population, the probability of selecting a sample that shows a t statistic from this sample or more extreme is x

c) If there is truly a difference in the population, the probability of selecting a sample that shows a t statistic from this sample or more extreme is x

d) If there is truly a difference in the sample, the probability of selecting a sample that shows a t statistic from this sample or more extreme is x

Answer 43

b) If there is truly no difference in the population, the probability of selecting a sample that shows a t statistic from this sample or more extreme is x