Related and unrelated t-tests | AQA A-Level Psychology Revision
- Revision Notes
- Aug 5
- 18 min read
Updated: 5 days ago
For 7182 specification, first teach in September 2025
AQA A-Level Psychology | Free Revision Notes
Estimated study time: 55 minutes
This Related and unrelated t-tests A-Level Psychology revision page explains how psychologists select a parametric test for a difference using interval data. You will learn that a related t-test is used when two sets of scores are paired, while an unrelated t-test is used when they come from independent groups. You will also distinguish both t-tests from non-parametric tests of difference. This lesson builds on choosing an inferential test and experimental designs.
Learning Objectives 🎯
By the end of this revision page, you should be able to:
Identify when a related t-test should be used.
Identify when an unrelated t-test should be used.
Distinguish related from unrelated data.
Recognise interval data in an unfamiliar study.
Distinguish parametric t-tests from non-parametric tests of difference.
Select and justify the correct test from a psychological research scenario.
Revision Notes 📚
Related and unrelated t-tests in A-Level Psychology
The related and unrelated t-tests are inferential statistical tests used to investigate a difference.
Both tests require data measured at the interval level.
The difference between them concerns the relationship between the two sets of scores:
a related t-test is used with related data;
an unrelated t-test is used with unrelated data.
The central selection rules are:
$$\text{Difference}+\text{related data}+\text{interval data}=\text{Related }t\text{-test}$$
$$\text{Difference}+\text{unrelated data}+\text{interval data}=\text{Unrelated }t\text{-test}$$
The researcher must therefore identify:
The purpose of the analysis.
The experimental design.
The level of measurement.
Both t-tests investigate a difference
A test of difference examines whether two conditions, groups or occasions produce different results.
Examples include investigating whether:
memory scores differ between silence and background-noise conditions;
response times differ before and after training;
two groups produce different concentration scores;
performance differs between two treatments.
Words suggesting a test of difference include:
difference;
differs;
higher;
lower;
more;
fewer;
increase;
decrease;
effect;
compare.
Example hypothesis
A psychologist might predict:
There will be a difference in the time taken to complete a task before and after practice.
This is a test of difference because the researcher compares performance on two occasions.
The researcher must then identify:
whether the scores are related or unrelated;
whether the data are measured at the interval level.
When neither t-test is appropriate
The related and unrelated t-tests should not be selected when the study investigates:
a correlation between two co-variables;
an association between categories;
nominal data;
ordinal data.
For example:
There will be a correlation between hours of sleep and memory-test score.
This is a test of correlation rather than a test of difference. If the data are interval, Pearson’s \(r\) would be appropriate.
Correlation tests are covered in Spearman’s rho and Pearson’s r.
Interval data
What are interval data?
Interval data consist of numerical measurements in which the intervals between values are equal.
This means that the difference between one pair of adjacent values is the same size as the difference between another pair.
Examples include:
time measured in seconds;
response time measured in milliseconds;
distance measured in centimetres;
the number of words recalled;
numerical test scores measured using consistent units.
Equal intervals
Suppose a psychologist measures task-completion time in seconds.
The difference between:
$$12\text{ seconds and }13\text{ seconds}$$
is the same size as the difference between:
$$18\text{ seconds and }19\text{ seconds}$$
In both cases, the difference is:
$$1\text{ second}$$
These are interval data.
Worked example: memory scores
A researcher records the number of words recalled by participants:
$$8,\ 10,\ 12,\ 15,\ 17$$
The difference between:
$$8\text{ and }10$$
is:
$$2\text{ words}$$
The difference between:
$$15\text{ and }17$$
is also:
$$2\text{ words}$$
The scores are numerical and use consistent units, so they may be treated as interval data.
Recognising interval data
Look for measurements such as:
seconds;
milliseconds;
centimetres;
number of correct answers;
numerical scores measured using equal units.
Do not assume that data are interval merely because numbers have been used.
Numbers may represent:
nominal category codes;
ordinal ranks;
ordered ratings.
The distinction is explained fully in levels of measurement.
Rating scales are not automatically interval
Suppose participants rate anxiety using:
$$1=\text{very low anxiety}$$
to:
$$5=\text{very high anxiety}$$
The values have a meaningful order, but the psychological difference between:
$$1\text{ and }2$$
cannot automatically be assumed to equal the difference between:
$$4\text{ and }5$$
These ratings are ordinal rather than interval.
A non-parametric test of difference would therefore normally be selected instead of a t-test.
Related and unrelated data
Why experimental design matters
Both t-tests have the same:
purpose;
level of measurement.
They are both tests of difference using interval data.
The experimental design determines whether the related or unrelated version should be selected.
Related data
Related data contain scores that are meaningfully paired.
Related scores are produced by:
repeated measures designs;
matched pairs designs.
Each score in one condition has a corresponding score in the other condition.
Repeated measures design
In a repeated measures design, the same participants take part in both conditions.
For example, each participant completes a memory test:
before receiving training;
after receiving training.
Each participant’s before score is paired with their own after score.
The data are related because the same people provide both sets of results.
Example of related data
Participant | Words recalled before training | Words recalled after training |
A | \(8\) | \(12\) |
B | \(11\) | \(14\) |
C | \(10\) | \(13\) |
D | \(14\) | \(16\) |
E | \(9\) | \(12\) |
Participant A’s pair of scores is:
$$(8,12)$$
Participant B’s pair is:
$$(11,14)$$
The pairing must be retained during the analysis.
Matched pairs design
In a matched pairs design, each participant in one condition is paired with a similar participant in another condition.
Participants may be matched according to:
age;
ability;
an initial test score;
another relevant characteristic.
Although different participants complete the two conditions, their scores are related because each person has been deliberately matched with another.
Example of matched-pairs data
A psychologist compares two memory techniques.
Participants are matched according to their initial memory score. One member of each pair uses Technique A, while the other uses Technique B.
The researcher records the number of words recalled.
The study produces:
a test of difference;
related data through matching;
interval memory scores.
The appropriate test is:
$$\boxed{\text{Related }t\text{-test}}$$
Unrelated data
Unrelated data come from separate participants or groups whose scores are not meaningfully paired.
An independent groups design normally produces unrelated data.
For example:
Group A completes a memory task in silence.
Group B completes the task with background noise.
Each participant takes part in only one condition.
There is no direct pairing between a score in Group A and a score in Group B.
Example of unrelated data
Silence group | Memory score | Background-noise group | Memory score |
A | \(16\) | F | \(11\) |
B | \(15\) | G | \(13\) |
C | \(18\) | H | \(12\) |
D | \(14\) | I | \(10\) |
E | \(17\) | J | \(14\) |
The scores come from separate groups.
The data are therefore unrelated.
Related and unrelated data compared
Feature | Related data | Unrelated data |
Same participants complete both conditions | Yes, in repeated measures | No |
Participants are deliberately matched | Possibly | No |
Separate independent groups | No | Yes |
Each score has a paired score | Yes | No |
Interval difference test | Related t-test | Unrelated t-test |
The related t-test
When should a related t-test be used?
A related t-test should be used when:
The researcher is testing for a difference.
The two sets of scores are related.
The data are measured at the interval level.
The selection rule is:
$$\text{Difference}+\text{related}+\text{interval}=\text{Related }t\text{-test}$$
Suitable experimental designs
A related t-test may be used with:
repeated measures designs;
matched pairs designs.
Both designs produce paired scores.
Worked example: repeated measures
A psychologist investigates whether practice changes participants’ response times.
Each participant completes the same task:
before practice;
after practice.
Response time is measured in milliseconds.
Purpose
The researcher compares performance on two occasions:
$$\text{Test of difference}$$
Design
The same participants provide both sets of scores:
$$\text{Related data}$$
Level of measurement
Response time is measured in milliseconds:
$$\text{Interval data}$$
Appropriate test
$$\boxed{\text{Related }t\text{-test}}$$
Complete justification
A suitable explanation would be:
The related t-test is appropriate because the researcher is testing for a difference, the same participants complete the task before and after practice so the scores are related, and response time is measured at the interval level.
Worked example: matched pairs
A researcher compares two teaching methods.
Participants are matched according to their score on an initial assessment. One member of each pair receives Method A and the other receives Method B.
The researcher records each participant’s score on a final test.
Purpose
The two methods are compared:
$$\text{Test of difference}$$
Design
Participants are matched:
$$\text{Related data}$$
Level of measurement
The final test produces numerical scores using equal units:
$$\text{Interval data}$$
Appropriate test
$$\boxed{\text{Related }t\text{-test}}$$
The related t-test is not selected from design alone
A repeated measures or matched pairs design establishes that the data are related.
However, the researcher must still identify the level of measurement.
For related tests of difference:
Level of measurement | Appropriate test |
Nominal signs | Sign test |
Ordinal | Wilcoxon |
Interval | Related t-test |
Therefore, repeated measures does not automatically mean that a related t-test should be used.
The unrelated t-test
When should an unrelated t-test be used?
An unrelated t-test should be used when:
The researcher is testing for a difference.
The two sets of scores are unrelated.
The data are measured at the interval level.
The selection rule is:
$$\text{Difference}+\text{unrelated}+\text{interval}=\text{Unrelated }t\text{-test}$$
Suitable experimental design
An unrelated t-test is used with an independent groups design.
Different participants take part in each condition.
Worked example: independent groups
A psychologist investigates whether sleep deprivation affects concentration.
Group A sleeps normally before completing a concentration task.
Group B is sleep deprived before completing the task.
The researcher records each participant’s numerical concentration score.
Purpose
The researcher compares two conditions:
$$\text{Test of difference}$$
Design
Different participants complete the conditions:
$$\text{Unrelated data}$$
Level of measurement
The concentration results are numerical scores measured using equal units:
$$\text{Interval data}$$
Appropriate test
$$\boxed{\text{Unrelated }t\text{-test}}$$
Complete justification
A suitable explanation would be:
The unrelated t-test is appropriate because the researcher is testing for a difference, different participants take part in the two conditions so the scores are unrelated, and the concentration scores are measured at the interval level.
Worked example: two naturally occurring groups
A researcher compares the time taken to solve a problem by:
participants who regularly play strategy games;
participants who do not regularly play strategy games.
Different participants belong to each group.
Time is measured in seconds.
Purpose
The researcher compares two groups:
$$\text{Test of difference}$$
Design
The groups contain different participants:
$$\text{Unrelated data}$$
Level of measurement
Time is measured using equal units:
$$\text{Interval data}$$
Appropriate test
$$\boxed{\text{Unrelated }t\text{-test}}$$
The unrelated t-test is not selected from design alone
An independent groups design establishes that the data are unrelated.
The level of measurement must still be considered.
For unrelated tests of difference:
Level of measurement | Appropriate test |
Ordinal | Mann-Whitney |
Interval | Unrelated t-test |
Different participants do not automatically mean that an unrelated t-test is appropriate.
Comparing the related and unrelated t-tests
Similarities
The related and unrelated t-tests are similar because both:
are inferential statistical tests;
test for a difference;
are used with interval data;
produce an observed test value;
require comparison with a critical value;
may be used to determine statistical significance;
lead to a decision about the null hypothesis;
are parametric tests.
Differences
The main difference is the relationship between the two sets of scores.
Feature | Related t-test | Unrelated t-test |
Purpose | Difference | Difference |
Level of measurement | Interval | Interval |
Relationship between scores | Related | Unrelated |
Suitable design | Repeated measures or matched pairs | Independent groups |
Direct pairing | Each score has a corresponding score | Scores are not paired |
Central distinction
Remember:
$$\text{Related }t\text{-test}=\text{Related interval difference}$$
$$\text{Unrelated }t\text{-test}=\text{Unrelated interval difference}$$
Parametric and non-parametric tests
What is a parametric test?
The related and unrelated t-tests are parametric tests.
Within A-Level Psychology test selection, they are distinguished from non-parametric tests because they are used with interval data.
The appropriate t-test is then selected according to whether the scores are related or
unrelated.
What is a non-parametric test?
The non-parametric tests of difference included in the specification are:
the sign test;
Wilcoxon;
Mann-Whitney.
These tests use nominal or ordinal data rather than interval data.
Parametric and non-parametric tests compared
Feature | Parametric t-tests | Non-parametric tests of difference |
Purpose | Test of difference | Test of difference |
Data used in test selection | Interval | Nominal or ordinal |
Related-data options | Related t-test | Sign test or Wilcoxon |
Unrelated-data option | Unrelated t-test | Mann-Whitney |
Example data | Time in seconds | Ranks or ordered ratings |
Related tests of difference
For related data, the level of measurement distinguishes the tests.
Nominal data
$$\text{Sign test}$$
Ordinal data
$$\text{Wilcoxon}$$
Interval data
$$\text{Related }t\text{-test}$$
Unrelated tests of difference
For unrelated data, the level of measurement distinguishes the tests.
Ordinal data
$$\text{Mann-Whitney}$$
Interval data
$$\text{Unrelated }t\text{-test}$$
Related t-test and Wilcoxon compared
Both tests are used for:
a test of difference;
related scores.
The difference is the level of measurement.
Feature | Related t-test | Wilcoxon |
Purpose | Difference | Difference |
Data relationship | Related | Related |
Level of measurement | Interval | Ordinal |
Type | Parametric | Non-parametric |
Example | Response times in seconds | Ordered anxiety ratings |
The Wilcoxon test is covered in Wilcoxon and Mann-Whitney tests.
Unrelated t-test and Mann-Whitney compared
Both tests are used for:
a test of difference;
unrelated scores.
The difference is the level of measurement.
Feature | Unrelated t-test | Mann-Whitney |
Purpose | Difference | Difference |
Data relationship | Unrelated | Unrelated |
Level of measurement | Interval | Ordinal |
Type | Parametric | Non-parametric |
Example | Numerical memory scores | Ranked memory performance |
Related t-test and the sign test compared
Both can be used for a difference involving related data.
Feature | Related t-test | Sign test |
Purpose | Difference | Difference |
Data relationship | Related | Related |
Level of measurement | Interval | Nominal signs |
Information retained | Numerical scores | Direction of each difference |
Type | Parametric | Non-parametric |
The sign test is covered in the sign test.
Why the research purpose must be identified first
Interval data do not automatically require a t-test.
If a researcher investigates a correlation between two interval co-variables, the appropriate test is:
$$\text{Pearson's }r$$
If a researcher compares two sets of interval scores, a t-test may be appropriate.
Therefore:
interval correlation requires Pearson’s \(r\);
related interval difference requires a related t-test;
unrelated interval difference requires an unrelated t-test.
Selecting the correct test from a scenario
The three-question method
Use the following questions.
Question 1: Is the researcher testing a difference?
If the researcher compares conditions, groups or occasions, the study is probably testing a difference.
If the researcher investigates a relationship between two co-variables, neither t-test is appropriate.
Question 2: Are the scores related or unrelated?
Related scores come from:
repeated measures;
matched pairs.
Unrelated scores come from:
independent groups.
Question 3: Are the data interval?
Look for numerical measurements with equal units.
If the data are ordinal, select Wilcoxon or Mann-Whitney instead.
Decision process
Related interval data
$$\boxed{\text{Related }t\text{-test}}$$
Unrelated interval data
$$\boxed{\text{Unrelated }t\text{-test}}$$
Related ordinal data
$$\boxed{\text{Wilcoxon}}$$
Unrelated ordinal data
$$\boxed{\text{Mann-Whitney}}$$
Related nominal signs
$$\boxed{\text{Sign test}}$$
Worked scenario 1
The same participants complete a memory task before and after a revision programme.
The researcher records the number of words recalled.
Purpose
Two occasions are compared:
$$\text{Difference}$$
Design
The same participants provide both scores:
$$\text{Related}$$
Level of measurement
The number of words recalled is a numerical score using consistent units:
$$\text{Interval}$$
Correct test
$$\boxed{\text{Related }t\text{-test}}$$
Worked scenario 2
One group completes a concentration task in silence. A different group completes the task while listening to speech.
The researcher records numerical concentration scores.
Purpose
Two conditions are compared:
$$\text{Difference}$$
Design
Different participants complete the conditions:
$$\text{Unrelated}$$
Level of measurement
The concentration scores use equal numerical units:
$$\text{Interval}$$
Correct test
$$\boxed{\text{Unrelated }t\text{-test}}$$
Worked scenario 3
The same participants rate anxiety before and after relaxation using an ordered scale.
Purpose
$$\text{Difference}$$
Design
$$\text{Related}$$
Level of measurement
$$\text{Ordinal}$$
Correct test
$$\boxed{\text{Wilcoxon}}$$
The related t-test is not appropriate because the data are ordinal rather than interval.
Worked scenario 4
Two independent groups rate the usefulness of two treatments from very unhelpful to very helpful.
Purpose
$$\text{Difference}$$
Design
$$\text{Unrelated}$$
Level of measurement
$$\text{Ordinal}$$
Correct test
$$\boxed{\text{Mann-Whitney}}$$
The unrelated t-test is not appropriate because the results are ordered ratings.
Worked scenario 5
Participants’ scores are classified as having increased or decreased after an intervention.
Purpose
$$\text{Difference}$$
Design
$$\text{Related}$$
Level of measurement
The directions are converted into categories:
$$\text{Nominal}$$
Correct test
$$\boxed{\text{Sign test}}$$
A related t-test would not be appropriate because the test uses nominal signs rather than interval scores.
Worked scenario 6
A researcher investigates whether hours of sleep are related to response time in milliseconds.
Purpose
The researcher investigates two co-variables:
$$\text{Correlation}$$
Level of measurement
Both co-variables are interval.
Correct test
$$\boxed{\text{Pearson's }r}$$
Neither t-test is appropriate because the researcher is not comparing two conditions or groups.
Interpreting statistical significance
Observed values
A t-test produces an observed value calculated from the research data.
This is commonly represented using:
$$t$$
The observed value is compared with a critical value obtained from the relevant statistical table.
Critical values
The correct critical value depends on information including:
the t-test being used;
the sample information required by the table;
the selected significance level;
whether the hypothesis is directional or non-directional.
The use of critical values is covered in probability and significance.
Significance rule for t-tests
For related and unrelated t-tests, the result is statistically significant when the observed value is equal to or greater than the critical value.
The decision rule is:
$$t_{\text{observed}}\geq t_{\text{critical}}$$
A larger observed value provides stronger evidence against the null hypothesis.
Worked significant result
A related t-test produces:
$$t_{\text{observed}}=2.48$$
The critical value is:
$$t_{\text{critical}}=2.20$$
Compare the values:
$$2.48\geq2.20$$
The result is statistically significant.
The null hypothesis should be rejected.
Worked non-significant result
An unrelated t-test produces:
$$t_{\text{observed}}=1.76$$
The critical value is:
$$t_{\text{critical}}=2.10$$
Compare the values:
$$1.76<2.10$$
The result is not statistically significant.
The null hypothesis should be retained.
Equality reaches the threshold
Suppose:
$$t_{\text{observed}}=2.15$$
and:
$$t_{\text{critical}}=2.15$$
Because:
$$2.15\geq2.15$$
the result is statistically significant.
The observed value has reached the critical threshold.
Writing a complete conclusion
A complete statistical conclusion should include:
The observed value.
The critical value.
The comparison between them.
Whether the result is statistically significant.
Whether the null hypothesis is rejected or retained.
A conclusion referring to the conditions or groups.
For example:
The observed t-value of \(2.48\) was greater than the critical value of \(2.20\). The result was statistically significant at \(p\leq0.05\), so the null hypothesis was rejected. There was a statistically significant difference in memory performance before and after the revision programme.
A non-significant conclusion
The observed t-value of \(1.76\) was lower than the critical value of \(2.10\). The result was not statistically significant at the \(0.05\) level, so \(p>0.05\). The null hypothesis was retained. There was insufficient evidence of a difference between the two conditions.
Statistical significance and study quality
A statistically significant t-test result does not automatically show that:
the effect is large;
the research has high validity;
the sample is representative;
the method was free from bias;
the result has practical importance.
Inferential testing provides a probability-based decision. The quality of the research procedure must still be evaluated separately.
Writing a strong test justification
Related t-test structure
A related t-test is appropriate because the researcher is testing for a difference, the same participants complete both conditions so the data are related, and the scores are measured at the interval level.
For matched pairs:
A related t-test is appropriate because the researcher is testing for a difference, the participants are matched so their scores are related, and the outcome measure produces interval data.
Unrelated t-test structure
An unrelated t-test is appropriate because the researcher is testing for a difference, different participants complete the two conditions so the data are unrelated, and the scores are measured at the interval level.
Why short justifications may lose marks
An answer such as:
Use a related t-test because the same participants were used.
does not identify:
that the study tests a difference;
that the data are interval.
An answer such as:
Use an unrelated t-test because the results are interval.
does not identify:
the purpose of the analysis;
that the groups are unrelated.
For a full explanation, include all three selection factors.
Complete test-selection comparison
Statistical test | Purpose | Data relationship | Level of measurement |
Sign test | Difference | Related | Nominal |
Wilcoxon | Difference | Related | Ordinal |
Mann-Whitney | Difference | Unrelated | Ordinal |
Related t-test | Difference | Related | Interval |
Unrelated t-test | Difference | Unrelated | Interval |
Spearman’s rho | Correlation | Paired co-variable scores | Ordinal |
Pearson’s \(r\) | Correlation | Paired co-variable scores | Interval |
Chi-squared | Association | Categorical frequencies | Nominal |
Final test-selection checklist
Before choosing a t-test, ask:
Research purpose
Is the researcher comparing conditions, groups or occasions?
Is the study testing a difference rather than a correlation or association?
Experimental design
Do the same participants complete both conditions?
Have participants been matched?
Are separate independent groups used?
Are the scores related or unrelated?
Level of measurement
Are the scores numerical measurements?
Do equal differences represent equal units?
Are the values genuinely interval rather than ranks or rating categories?
Final decision
Related interval difference means related t-test.
Unrelated interval difference means unrelated t-test.
Ordinal data require a non-parametric alternative.
Key Words 🔑
Key word | Student-friendly definition | How it may be used in an exam |
Related t-test | A parametric inferential test used for a difference between two sets of related interval data. | You may select it for repeated measures or matched pairs interval scores. |
Unrelated t-test | A parametric inferential test used for a difference between two sets of unrelated interval data. | You may select it for independent groups interval scores. |
Test of difference | An analysis examining whether two conditions, groups or occasions produce different results. | Both t-tests are tests of difference. |
Related data | Scores paired through the same participants or deliberately matched participants. | Related interval data require a related t-test. |
Unrelated data | Scores obtained from independent participants or groups. | Unrelated interval data require an unrelated t-test. |
Repeated measures | A design in which the same participants take part in both conditions. | It produces related data. |
Matched pairs | A design in which participants are paired according to relevant characteristics. | It also produces related data. |
Independent groups | A design in which different participants take part in each condition. | It produces unrelated data. |
Interval data | Numerical data measured using equal intervals or units. | Both t-tests require interval data. |
Parametric test | A statistical test associated here with interval-level data. | Related and unrelated t-tests are parametric tests. |
Non-parametric test | A statistical test used here with nominal or ordinal data. | Sign, Wilcoxon and Mann-Whitney are non-parametric tests of difference. |
Observed value | The test statistic calculated from the study’s results. | It is compared with a critical value. |
Critical value | The statistical-table threshold used to determine significance. | For a t-test, the observed value must meet or exceed it. |
Statistical significance | A judgement that a result meets the selected probability criterion. | A significant result leads to rejection of the null hypothesis. |
Null hypothesis | A prediction that there is no difference and that any apparent result is due to chance. | It is rejected when the t-test result is significant. |
Directional hypothesis | A hypothesis predicting the direction of a difference. | It requires a one-tailed critical value. |
Non-directional hypothesis | A hypothesis predicting a difference without stating its direction. | It requires a two-tailed critical value. |
Hints from the Examiner Reports 💡
No lesson-specific examiner guidance was identified in the provided reports.
Common Mistakes ⚠️
Mistake: Selecting a t-test whenever the results contain numbers.
Why this is incorrect:
Numbers may represent ranks, ratings or category codes rather than interval measurements.
How to improve:
Identify what the values mean and whether the intervals between them are equal.
Mistake: Selecting a related t-test for every repeated measures study.
Why this is incorrect:
Repeated measures establishes related data, but the results must also be interval.
How to improve:
Use:
$$\text{Difference}+\text{related}+\text{interval}$$
Mistake: Selecting an unrelated t-test for every independent groups study.
Why this is incorrect:
An independent groups design establishes unrelated data, but an unrelated t-test also requires interval measurement.
How to improve:
If the data are ordinal, select Mann-Whitney instead.
Mistake: Treating matched-pairs data as unrelated.
Why this is incorrect:
Participants have been deliberately paired, so each score has a corresponding matched score.
How to improve:
Classify repeated measures and matched pairs as related designs.
Mistake: Treating ordered ratings as interval data.
Why this is incorrect:
The categories are ordered, but equal psychological differences between adjacent values cannot automatically be assumed.
How to improve:
Use Wilcoxon for related ordinal ratings or Mann-Whitney for unrelated ordinal ratings.
Mistake: Selecting Pearson’s \(r\) for a comparison of two conditions.
Why this is incorrect:
Pearson’s \(r\) tests a correlation between two interval co-variables rather than a difference between conditions.
How to improve:
Identify whether the study investigates a relationship or a difference.
Mistake: Confusing the related and unrelated t-tests.
Why this is incorrect:
The names refer to the relationship between the two sets of scores.
How to improve:
Remember:
$$\text{Related scores}\rightarrow\text{Related }t\text{-test}$$
$$\text{Unrelated scores}\rightarrow\text{Unrelated }t\text{-test}$$
Mistake: Assuming that all t-tests are non-parametric.
Why this is incorrect:
Related and unrelated t-tests are parametric tests associated with interval data.
How to improve:
Distinguish them from the non-parametric sign, Wilcoxon and Mann-Whitney tests.
Mistake: Reversing the t-test significance rule.
Why this is incorrect:
For a t-test, the observed value must be equal to or greater than the critical value.
How to improve:
Remember:
$$t_{\text{observed}}\geq t_{\text{critical}}$$
Mistake: Treating equal observed and critical values as non-significant.
Why this is incorrect:
Equality means that the observed value has reached the critical threshold.
How to improve:
Include the equality symbol:
$$\geq$$
Mistake: Justifying a test using only the level of measurement.
Why this is incorrect:
Interval data could require Pearson’s \(r\), a related t-test or an unrelated t-test.
How to improve:
Identify:
Difference, correlation or association.
Related or unrelated data.
Level of measurement.
Mistake: Giving only the name of the test when asked to justify it.
Why this is incorrect:
A justification must connect the test to features of the study.
How to improve:
Refer directly to the research purpose, experimental design and level of measurement.
Exam-Style Questions ✍️
Question 1
State when a related t-test should be used.[3 marks]
Question 2
State when an unrelated t-test should be used.[3 marks]
Question 3
Explain one similarity and one difference between the related and unrelated t-tests.[4 marks]
Question 4
The same participants complete a memory task before and after a training programme. The researcher records the number of words correctly recalled.
Identify an appropriate inferential statistical test. Explain your answer.[4 marks]
Question 5
One group completes a reaction-time task after sleep deprivation. A separate group completes the task after a normal night’s sleep.
Response time is measured in milliseconds.
Identify and justify the appropriate statistical test.[4 marks]
Question 6
Participants are matched according to an initial concentration score. One member of each pair completes a task in silence, while the other completes it with background noise.
The researcher records a numerical concentration score using equal units.
Identify the appropriate statistical test. Explain your answer.[4 marks]
Question 7
The same participants rate their anxiety before and after a relaxation activity using an ordered scale.
A researcher proposes using a related t-test.
Explain why this test is inappropriate and identify the correct test.[4 marks]
Question 8
Two independent groups rate the usefulness of two treatments from very unhelpful to very helpful.
A researcher proposes using an unrelated t-test.
Explain why this test is inappropriate and identify the correct test.[4 marks]
Question 9
A psychologist investigates whether hours of sleep are related to numerical memory scores measured using equal units.
A student selects a related t-test because each participant provides two scores.
Explain why the student is incorrect and identify the appropriate test.[4 marks]
Question 10
For each scenario, identify the appropriate inferential statistical test.
a) A difference between related interval scores.[1 mark]
b) A difference between unrelated interval scores.[1 mark]
c) A difference between related ordinal scores.[1 mark]
d) A difference between unrelated ordinal scores.[1 mark]
e) A difference between related nominal signs.[1 mark]
f) A correlation between two interval co-variables.[1 mark]
Question 11
A related t-test produces:
$$t_{\text{observed}}=2.64$$
The critical value is:
$$t_{\text{critical}}=2.18$$
a) Determine whether the result is statistically significant.[1 mark]
b) Explain your answer using the observed and critical values.[2 marks]
c) State what should happen to the null hypothesis.[1 mark]
Question 12
An unrelated t-test produces:
$$t_{\text{observed}}=1.82$$
The critical value is:
$$t_{\text{critical}}=2.06$$
a) Determine whether the result is statistically significant.[1 mark]
b) Explain your answer.[2 marks]
c) State an appropriate probability conclusion at the \(0.05\) significance level.[1 mark]
Question 13
A psychologist compares the numerical memory scores of two independent groups.
The unrelated t-test produces:
$$t_{\text{observed}}=2.31$$
The critical value at:
$$p\leq0.05$$
is:
$$t_{\text{critical}}=2.31$$
Write a complete statistical conclusion. Your answer should:
compare the observed and critical values;
state whether the result is statistically significant;
state whether the null hypothesis should be rejected or retained;
refer to the memory scores of the two groups.
[4 marks]
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