Chapter 9: Paired-Samples t Test

Question 1

Chapter 8 Review: When should you use a One-Sample t Test?

A: A one-sample t test is used to compare the means of two independent samples, assessing whether the difference between the two means is statistically significant.

B: To test the difference between a sample mean and
a known or hypothesized value of the mean in the population. To test whether the known or hypothesized value of the mean is true and supported by the sample mean.

C: In a one-sample t test, the goal is to determine if the sample mean is within a specific range of values, without consideration for a known or hypothesized population mean.

Answer 1

B: To test the difference between a sample mean and
a known or hypothesized value of the mean in the population. To test whether the known or hypothesized value of the mean is true and supported by the sample mean.

Here are some examples of this:
> Example 1: A researcher concluded that the average birth weight for babies in the US is 3,410 grams (null hypothesis/population mean). Our goal is to test whether this researcher’s statement is true using the weights from 500 babies (sample).

> Example 2: A consumer group is investigating a producer of diet meals to examine if its prepackaged meals actually contain the advertised 6 ounces (null hypothesis/population mean) of protein in each package. This group collects data from 120 packages (sample).

> Example 3: A depression score of 100 (null hypothesis/population mean) in one questionnaire indicates severe depression. A researcher wants to know whether the patients in a hospital have a mean depression score of 100 and therefore he collects the scores from 25 patients (sample).

Question 2

Review: What is a Within-Group Research Design?

A: A within-group research design involves comparing two or more independent groups, each providing separate sets of data for analysis, without considering repeated measurements from the same participants.

B: A within-group research design (within-subjects or
repeated-measures design) produces two or more
scores from the same participant. A classic within-group design collects two or more repeated measurements from the same group of
individuals (e.g., pretest and posttest). Special case: the two scores can also come from the same dyad (not the same participant).

C: Within-group research design focuses exclusively on collecting data from a single time point, with no consideration for repeated measurements or pretest-posttest comparisons within the same group.

Answer 2

B: A within-group research design (within-subjects or
repeated-measures design) produces two or more
scores from the same participant. A classic within-group design collects two or more repeated measurements from the same group of
individuals (e.g., pretest and posttest). Special case: the two scores can also come from the same dyad (not the same participant).

For example:
> Intervention Application = Two scores are obtained from the same group of people before (pre-test/intervention) and after (post-test/intervention).

> OR… Experimental Application = The same group of people are exposed to two different experimental conditions (stimuli 1 and stimuli 2).

> OR… Developmental Application = The same group of people are followed over time, with the goal of examining change or development (baseline + follow-up).

> OR… Dyadic Application = One score is obtained from each member of a natural dyad (e.g., romantic partners, siblings, peers). Dyad member #1 + dyad member #2.

YOU MIGHT WANT TO ADD THESE IMAGES TO YOUR CHEAT SHEET WITH THIS SIMPLIFIED DEFINITION (it’s slide 6-9).

Question 3

What is a paired-sample t test?

A: The paired-samples (dependent) t test is
appropriate for within-group designs with two
measurements. Two scores are first reduced to a single change or difference score, after which the procedure is identical to a one-sample t test.

B: A paired-sample t test is used to compare the means of two entirely unrelated groups, without any consideration for within-group designs or paired measurements.

C: The paired-sample t test is exclusively applied to between-group designs, where the focus is on comparing the means of two independent samples without any consideration for repeated measurements within the same group.

Answer 3

A: The paired-samples (dependent) t test is
appropriate for within-group designs with two
measurements. Two scores are first reduced to a single change or difference score, after which the procedure is identical to a one-sample t test.

The Difference Between One-Sample And Paired-Sample t Tests:
> One-sample t test: Han wants to know whether the population mean of orange sweetness is 5. She eats every orange and rates them.

> Paired-sample t test: Han wants to know whether
freezing oranges can change the sweetness level of
the oranges (the population mean of the change of
sweetness). For each orange, she first eats half of
the orange and rates it. After freezing the other half,
she eats and rates it again.

NOTE: Remember that in a paired-sample t test, each subject contributes two scores - a pre-test score and a post-test score.

Question 4

What are Difference (Change) Scores?

A: Difference scores, in the context of paired-samples t tests, represent the sum of the pre-test and post-test scores within each individual or dyad, providing a comprehensive measure of overall change.

B: The paired-samples t test requires two scores per pair (a pair can be two scores per individual or two scores per dyad). The hypotheses and computations involve difference scores (also called change scores). The difference score can be the difference between
post-test and pre-test within each individual or the
difference between a husband’s score and a wife’s score within each dyad.

C: In the paired-samples t test, difference scores refer to the average of the pre-test and post-test scores, emphasizing the total change observed across the entire sample without considering individual variations.

Answer 4

B: The paired-samples t test requires two scores per pair (a pair can be two scores per individual or two scores per dyad). The hypotheses and computations involve difference scores (also called change scores). The difference score can be the difference between
post-test and pre-test within each individual or the
difference between a husband’s score and a wife’s score within each dyad.

NOTE: When calculating the difference you can either do “pretest minus posttest” or “posttest minus pretest.” Just make sure you’re consistent and do the same thing across the board. Stick with your decision.

NOTE: Difference scores can be positive or negative,
but their meaning depends on how you subtract
the two scores.

RECOMMENDED: difference = post-test − pre-test. A positive difference score would then indicate
that scores increased from pre to post, and a
negative value indicates a decrease (THIS IS IMPORTANT. I WOULD PUT THIS ON YOUR CHEAT SHEET).

Question 5

What is the relationship between a one-sample t test and a paired-sample t test?

A: The one-sample t test and the paired-sample t test are entirely independent tests with no connection. While the one-sample t test assesses raw data, the paired-sample t test focuses exclusively on average differences between paired scores.

B: The paired-sample t test and the one-sample t test are essentially the same test with different names. The only distinction is that the paired-sample t test is used when working with continuous data, while the one-sample t test is employed for categorical data.

C: The paired-samples t test can be thought of as a variation of the one-sample t test. The difference between the two tests is that the paired-samples t test uses difference scores, while the one-sample t test uses raw data. In other words, the paired-samples t test is the one-sample t test applied to difference scores.

Answer 5

C: The paired-samples t test can be thought of as a variation of the one-sample t test. The difference between the two tests is that the paired-samples t test uses difference scores, while the one-sample t test uses raw data. In other words, the paired-samples t test is the one-sample t test applied to difference scores.

Question 6

The first thing in conducting a paired-samples t test is to compute the difference scores.

A. True
B. False

Answer 6

A. True

ASK ABOUT THIS!!!

Question 7

How would you apply the null hypothesis for within-group designs?

A: In within-group designs, the null hypothesis focuses on establishing a specific difference in the population means. It suggests that there is a predefined change or effect, and the population mean difference is expected to be a non-zero value.

B: Applying the null hypothesis to within-group designs involves assuming a population mean difference that aligns with the researcher’s expectations. It posits that a certain change or effect is present, rather than asserting a hypothesis of no difference.

C: The null hypothesis for within-group designs
targets the population mean difference. A typical application specifies a hypothesis of no difference (nothing happening, no change). The population mean of the difference scores is 0.

Answer 7

C: The null hypothesis for within-group designs
targets the population mean difference. A typical application specifies a hypothesis of no difference (nothing happening, no change). The population mean of the difference scores is 0.

Written like this: H0 : μDiff = 0

Question 8

How would you apply a one-tailed alternate hypothesis to a within-group design?

A: When applying a one-tailed alternate hypothesis to a within-group design, it means that both means are expected to show a statistically significant increase or decrease. The direction of the difference is not specified.

B: A one-tailed alternate hypothesis in a within-group design suggests that one mean is exactly equal to the other (≠ 0), allowing for the possibility of any type of difference between the two means, whether positive or negative.

C: A one-tailed hypothesis specifies that one mean is
higher than the other (> 0 or < 0).

Answer 8

C: A one-tailed hypothesis specifies that one mean is
higher than the other (> 0 or < 0).

Written like this if difference = post-test − pre-test, we assume scores increased from pre-test to post-test:
Ha : μDiff > 0

Written like this if we assume scores decrease from pre-test to post-test:
Ha : μDiff < 0

THIS IS SOMETHING YOU MIGHT WANT TO PUT ON YOUR CHEAT SHEET!!!

Question 9

How would you apply a two-tailed alternate hypothesis to a within-group design?

A: When applying a two-tailed alternate hypothesis to a within-group design, it means that there is no change expected in either direction. The hypothesis allows for the possibility of a stable, unchanging outcome.

B: A two-tailed hypothesis specifies that there is a
change. It could be a positive or negative change. Either way, something is happening in either direction.

C: A two-tailed alternate hypothesis in a within-group design specifies that the change could be in either direction (> 0 or < 0), but it doesn’t necessarily indicate the presence of any change. It merely allows for the exploration of potential differences.

Answer 9

B: A two-tailed hypothesis specifies that there is a
change. It could be a positive or negative change. Either way, something is happening in either direction.

Written like this: Ha : μDiff ≠ 0

Question 10

Standard error and paired-samples t test:

A: The standard error is the standard deviation of
the means (of the difference scores) from many
different random samples.

B: The standard error is the sampling error of
the means (of the difference scores) from many
different random samples.

C: The standard error is the deviation score of
the means (of the difference scores) from many
different random samples.

Answer 10

A: The standard error is the standard deviation of
the means (of the difference scores) from many
different random samples.

OR, you can say it like this: The average/expected difference between the sample mean of the difference score and the true population mean of the difference
score.

Written like this:
sx̄Diff = sDiff ÷ √N

Interpretation example:
N = 117
Standard deviation = 0.784
Standard error = .07
On average, a sample of size N = 117 would give an estimate that differs from the no-change hypothesis by ~ .07 (in standard error units)

PUT THIS ON YOUR CHEAT SHEET!!!

Question 11

Different from a one-sample t test, the standard error in a paired-samples t test is not a standard deviation of the sampling distribution.

A. True
B. False

Answer 11

B. False - you can’t change the definition of standard error

Question 12

t statistic and paired-samples t test:

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

B: The t statistic is a direct measure of the absolute difference between the estimate and hypothesis, providing an unstandardized value without converting it to a standardized metric.

C: In a paired-samples t test, the t statistic is only relevant when the estimate and hypothesis match perfectly. If there’s any deviation from the expected value, the t statistic loses its significance in assessing the difference between the two.

Answer 12

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

Written like this:
t = (x̄Diff - uDiff) ÷ (sDiff ÷ √N)

Which translates to:
Estimate vs. Hypothesis ÷ Standard error difference

Interpretation example:
t = 2.89
The difference between the sample mean and the hypothesis is nearly three times as large as the standard error

Question 13

Rephrasing the rule of thumb for paired-samples t test:

A: If the null hypothesis is true, 95% of all samples we could work with should have t statistics between ± C.V.

B: If the null hypothesis is accurate, we can expect approximately 95% of all samples to yield t statistics outside the range of ± C.V., indicating a lack of precision in the measurements.

C: Assuming the null hypothesis is invalid, around 95% of all samples should produce t statistics within the range of ± C.V., illustrating a high level of consistency and reliability in the measurements.

Answer 13

A: If the null hypothesis is true, 95% of all samples we could work with should have t statistics between ± C.V.

NOTE: We obtain critical values in Jamovi for paired-samples t test the same way we do for one-sample t test

YOU SHOULD PUT THIS SLIDE/IMAGE ON YOUR CHEAT SHEET!!!

Question 14

Please look at the image on slide 51. Based on the t statistic, how would you conclude the p-value?

A. p>0.05
B. p<=0.05

Answer 14

B. p<=0.05

Question 15

PUT SLIDES 55 AND 56 ON YOUR CHEAT SHEET WITH IMAGES AND FORMULAS THAT YOU USED ON QUIZ #8!

Answer 15

PUT SLIDES 55 AND 56 ON YOUR CHEAT SHEET WITH IMAGES AND FORMULAS THAT YOU USED ON QUIZ #8!

Question 16

Please look at the image on slide 55. Based on the two-tailed hypothesis p-value, can we reject the null hypothesis?

A. Reject
B. Fail to reject

Answer 16

A. Reject

Question 17

What is the calculation for a 95% confidence interval?

A: 95% C.I. = sample estimate ± (C.V. × standard deviation)

B: 95% C.I. = sample estimate ± (C.V. + standard error)

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

Answer 17

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

C.V. × standard error is the margin of error

Remember that in Jamovi the mean and standard error will say “difference” after them since this is a paired-sample t-test.

Interpretation example:
> Let’s say H0 = 0 and the calculated margin of error is .066 and .353
> We can say that we are 95% sure that the true population mean difference falls somewhere between .066 and .353. A population mean difference of zero is not in this range. If the null hypothesis is true, the probability of this sample data or more extreme data must be smaller than .05 (statistically “significant”).

And here’s a practical interpretation example:
> We have evidence against the null hypothesis (i.e.,
we have “proven it guilty”), so we adopt the
alternate hypothesis.
> Final conclusion: the population mean of the change
scores is not 0, so the intervention produced a change in attitudes toward vaccinations. The sample means tell us that attitudes have changed at post-test (two-tailed test).

Question 18

True or false: T-statistics, p-values, and confidence intervals are all examples of statistical significance?

A: True

B: False

Answer 18

A: True

Question 19

What is practical significance?

A: Whether meaningful on a practical level

B: Whether or not we can prove the null hypothesis true

C: Whether we found our data in a practical or non-practical way

Answer 19

A: Whether meaningful on a practical level

Question 20

What is an example of practical significance?

A: T-statistics

B: Hypothesis testing

C: Effect size

Answer 20

C: Effect size

Question 21

Why would we use practical significance instead of statistical significance?

A: Practical significance is preferred over statistical significance because it allows researchers to disregard the impact of effect size. With statistical significance, the focus is solely on sample size, making it easier to draw meaningful conclusions.

B: When using practical significance instead of statistical significance, researchers can avoid considering the actual data distribution. Statistical significance is heavily influenced by the shape of the distribution, while practical significance provides a more straightforward assessment.

C: Statistical significance is impacted by sample size, whereas practical significance like effect size is independent of sample size. This matters because with a huge sample size, we almost always can reject the null hypothesis with statistical significance but this is not the case with practical significance.

Answer 21

C: Statistical significance is impacted by sample size, whereas practical significance like effect size is independent of sample size. This matters because with a huge sample size, we almost always can reject the null hypothesis with statistical significance but this is not the case with practical significance.

Question 22

So what is effect size?

A: Effect size measures the absolute difference between two group means, providing a straightforward indication of the magnitude of the observed change without standardization.

B: The standardized mean difference effect size
(Cohen’s d) characterizes the magnitude of the
change on a standardized metric.

C: In effect size calculations, the focus is on the direction of the change rather than its magnitude. Effect size captures whether the change is positive or negative, with no consideration for the standardized metric.

Answer 22

B: The standardized mean difference effect size
(Cohen’s d) characterizes the magnitude of the
change on a standardized metric.

Formula:
d = x̄2 - x̄1 ÷ sDiff

Which translates to the mean difference ÷ the standard deviation

It’s reported in standard deviation units

And here’s an interpretation example:
> On the 6-point scale of the data, the raw score
change between the two measures (x̄2 - x̄1) was .21

On a standardized metric, Cohen’s d equates this change to .267 (a little more than ¼) standard deviation units. This is in the small effect size range.

Question 23

Make sure the effect size guidelines image on slide 70 is on your cheat sheet!!!

Answer 23

Make sure the effect size guidelines image on slide 70 is on your cheat sheet!!!

Question 24

You will need to know the APA-Style Summary section-by-section for the final exam!!! It’s likely she will provide one and ask if it’s accurate or not:

Answer 24

Part 1: “We used a paired-samples t-test to examine the
change in vaccination attitudes scores from pretest to
posttest.”
> This part describes the type of test that was used as well as what the research question was.

Part 2: “Table 1 gives the descriptive statistics.”
> This part mentions the descriptive statistics

Part 3: “The analysis revealed a statistically significant
improvement”
> This part talks about whether the analysis was reviewed as statistically significant or not.

Part 4: “t(116) = 2.89, p = .005, and the 95%
confidence interval for the mean difference ranged
from .07 to .35.”
> This part provides the test statistic
> (116) represents the degrees of freedom

Part 5: “Finally, the standardized mean difference was just above Cohen’s small effect size benchmark (d = .26), indicating that the intervention produced a relatively small improvement.”
> This part provides the practical significance by reporting the effect size.

Question 25

I would add the two-tailed probability interpretation images from slides 78, 83, and 84 to your cheat sheet!!!

Answer 25

I would add the two-tailed probability interpretation images from slides 78, 83, and 84 to your cheat sheet!!!

Question 26

I would add the one-tailed probability interpretation images from slides 79, 81, 82, 83, and 84 to your cheat sheet!!!

Answer 26

I would add the one-tailed probability interpretation images from slides 79, 81, 82, 83, and 84 to your cheat sheet!!!

Question 27

Jamovi paired-sample t test:

Yes or no: Is the mean difference equal to the difference between the post-test mean and the pre-test mean (post-test mean minus pre-test mean)?

A: Yes

B: No

Answer 27

A: Yes

PUT THIS ON YOUR CHEAT SHEET!

Question 28

Jamovi paired-sample t test:

Yes or no: Can we determine whether or not we can reject or fail to reject the null hypothesis by comparing the t-statistic and the critical value?

A: Yes

B: No

Answer 28

A: Yes

Example: Let’s say our t-statistic is 2.89 and our critical values are -1.981 and 1.981. Since the absolute value of our t-statistic (2.89) is greater than our CV (1.981) we can reject the null hypothesis (it falls outside the range of the critical value).

PUT THIS ON YOUR CHEAT SHEET!

Question 29

Jamovi paired-sample t test:

Yes or no: Can we determine whether or not we can reject or fail to reject the null hypothesis by comparing our null hypothesis to our confidence intervals?

A: Yes

B: No

Answer 29

A: Yes

For example: If our null hypothesis is 0 and our confidence intervals are 0.0659 and 0.353 we can reject the null hypothesis because 0 is not within our interval range.

PUT THIS ON YOUR CHEAT SHEET!

Question 30

Computing Scale Scores in Jamovi - PUT THIS ON YOUR CHEAT SHEET!!!

Answer 30

Step 1: Create a new blank column and select the “New Computed Variable” box

Step 2: Name your variable

Step 3: In the functions box (fx), scroll down and double-click on sum. “SUM () will appear to the right of the =

Step 4: Double-click on all the variables you wish to sum under the “variables” box and make sure commas separate each item

Step 5: Hit enter and notice that a new column with scale scores will appear

Question 31

Computing Mean Difference, Standard Error Difference, T-Statistic, P-Value, Confidence Interval, and Cohen’s d Effect Size in Jamovi - PUT THIS ON YOUR CHEAT SHEET!!!

Answer 31

Step 1: Click Analyses, T-Tests, then Paired Samples T-Test

Step 2: Drag over your variables in the order that you want to subtract them (ex: post-test minus pre-test)

Step 3: Select whether the hypothesis is >, <, or ≠

Step 4: Select Mean Difference, Confidence Interval, Effect Size & Descriptives

Question 32

Computing Critical Value in Jamovi - PUT THIS ON YOUR CHEAT SHEET!!!

Answer 32

Step 1: Click distrACTION and then T-Distribution

Step 2: Enter the Degrees of Freedom (N-1)

Step 3: Click Compute Quantile(s) and Central Interval Quantiles, and change p = to 0.95

Step 4: Hit enter

NOTE: If the absolute value of your T-Statistic is greater or less than your Critical Value you can reject the null hypothesis

Question 33

FOR YOUR CHEAT SHEET:

Margin of error formula

Answer 33

Critical Value × Standard Error

Question 34

Assignment 4: This is basically just interpreting standard error. The values from the assignment don’t really matter. It’s the interpretation that matters.

How do we interpret the standard error of 0.603? Check all that apply.

a) On average, a sample of 62 participants should produce a sample mean change that differs from the hypothesized population mean change by 0.603 points

b) If we conduct the study with the same sample size many times, the standard deviation of the sample means of the attitude change scores from those studies is 0.603 points

c) The average difference between a parent’s vaccine attitude scale change score and the sample mean of the change scores is 0.603 points

d) The expected difference between the sample mean of the attitude change scores and the true population mean change is 0.603 points

Answer 34

a) On average, a sample of 62 participants should produce a sample mean change that differs from the hypothesized population mean change by 0.603 points

b) If we conduct the study with the same sample size many times, the standard deviation of the sample means of the attitude change scores from those studies is 0.603 points

d) The expected difference between the sample mean of the attitude change scores and the true population mean change is 0.603 points

This is basically just interpreting standard error. The values from the assignment don’t really matter. It’s the interpretation that matters.

Question 35

Assignment 4: This is basically just interpreting a t-statistic. The values from the assignment don’t really matter. It’s the interpretation that matters.

The t-statistic value is 10.5. Which of the following interpretations is correct for the t statistic?

a) The pre-test average was about 10.5 times smaller than the post-test average

b) The difference between pre-test and post-test was about 10.5 times larger than the difference you would expect due to random chance (i.e., sampling error)

c) The change between pre-test and post-test averages was equivalent to 10.5 sample standard deviation units

Answer 35

b) The difference between pre-test and post-test was about 10.5 times larger than the difference you would expect due to random chance (i.e., sampling error)

This is basically just interpreting a t-statistic. The values from the assignment don’t really matter. It’s the interpretation that matters.

Question 36

Assignment 4: This is basically just interpreting the p-value. The values from the assignment don’t really matter. It’s the interpretation that matters.

The probability value (p-value) for the t-test is smaller than .001. Which of the following interpretations is correct?

a) The probability of drawing a sample of 62 parents that produces the t-statistic in the current sample or a more extreme t-statistic is smaller than .001

b) If there is truly no change in the population, the probability of selecting a sample of 62 parents that shows the t-statistic in the current sample or a more extreme t-statistic is smaller than .001

c) The probability that the hypothesis of no change is true is smaller than .001

Answer 36

b) If there is truly no change in the population, the probability of selecting a sample of 62 parents that shows the t-statistic in the current sample or a more extreme t-statistic is smaller than .001

This is basically just interpreting the p-value. The values from the assignment don’t really matter. It’s the interpretation that matters

Question 37

Assignment 4: This is basically just interpreting the confidence interval. The values from the assignment don’t really matter. It’s the interpretation that matters.

Which of the following interpretations is correct for the confidence interval?

a) We are 95% confident that the range 5.13 to 7.54 includes the true population mean difference of vaccine attitudes in the United States if the entire population read the “read scientific research” materials.

b) 95% of all random samples of participants reading the “read scientific research” materials would yield the range 5.13 to 7.54.

c) The true population mean difference of vaccine attitudes in the United States must fall within the range 5.13 to 7.54 if the entire population read the “read scientific research” materials.

Answer 37

a) We are 95% confident that the range 5.13 to 7.54 includes the true population mean difference of vaccine attitudes in the United States if the entire population read the “read scientific research” materials

This is basically just interpreting the confidence interval. The values from the assignment don’t really matter. It’s the interpretation that matters.