Probability and significance | AQA A-Level Psychology Revision
- Revision Notes
- Aug 5
- 16 min read
Updated: 6 days ago
For 7182 specification, first teach in September 2025
AQA A-Level Psychology | Free Revision Notes
Estimated study time: 60 minutes
This Probability and significance A-Level Psychology revision page explains how psychologists judge whether a research finding is likely to have occurred through chance. You will learn how probability levels are expressed, how significance thresholds are selected, and how observed and critical values are compared. The lesson builds on introduction to statistical testing and the sign test, preparing you to interpret the outcomes of the inferential tests required in AQA Research Methods.
Learning Objectives 🎯
By the end of this revision page, you should be able to:
Define probability and statistical significance.
Explain the meaning of the \(0.05\) and \(0.01\) significance levels.
Distinguish one-tailed and two-tailed significance decisions.
Select an appropriate critical value from a statistical table.
Compare observed and critical values using the correct decision rule.
Determine whether a result is statistically significant.
State statistical conclusions using appropriate symbols and psychological terminology.
Revision Notes 📚
Probability and significance in A-Level Psychology
Psychologists usually collect data from a sample rather than from every member of the target population.
A study may produce an apparent:
difference between two conditions;
difference between two groups;
correlation between two co-variables;
association between categorical variables.
However, a pattern found in a sample might have occurred through chance variation.
Probability and statistical significance allow the researcher to judge whether a result is sufficiently unlikely to have occurred by chance for the null hypothesis to be rejected.
The AQA specification requires students to understand:
probability and significance;
the use of statistical tables;
critical values;
the interpretation of statistical significance;
the use of appropriate statistical symbols.
What is probability?
Probability is the likelihood that an event or outcome will occur.
Probability is represented using:
$$p$$
A probability can range from:
$$0$$
to:
$$1$$
A probability of:
$$p=0$$
means that an outcome has no probability of occurring.
A probability of:
$$p=1$$
means that the outcome is certain.
A probability of:
$$p=0.50$$
represents:
$$50\%$$
Probability values can be converted into percentages using:
$$\text{Percentage probability}=p\times100$$
For example:
$$0.05\times100=5\%$$
and:
$$0.01\times100=1\%$$
Probability in inferential testing
In inferential testing, the researcher considers the probability of obtaining the observed result if the null hypothesis is true.
The null hypothesis states that there is no genuine difference, correlation or association, and that any apparent result is due to chance.
A small probability suggests that the observed pattern would be unusual if the null hypothesis were true.
The researcher therefore compares the result with a predetermined significance level.
What is a significance level?
A significance level is the probability threshold used to determine whether a research result is statistically significant.
The significance level is selected before the result is interpreted.
Common significance levels include:
$$p\leq0.05$$
and:
$$p\leq0.01$$
The symbol:
$$\leq$$
means less than or equal to.
Therefore:
$$p\leq0.05$$
means that the probability is less than or equal to \(0.05\).
The \(0.05\) significance level
The criterion:
$$p\leq0.05$$
means that the probability of obtaining the result through chance, if the null hypothesis is true, is no more than:
$$5\%$$
This may also be expressed as a confidence level of:
$$95\%$$
A result meeting this criterion is described as statistically significant at the \(0.05\) level.
For example:
The result was statistically significant at \(p\leq0.05\).
This means that the result met the selected \(5\%\) probability threshold.
The \(0.01\) significance level
The criterion:
$$p\leq0.01$$
means that the probability of obtaining the result through chance, if the null hypothesis is true, is no more than:
$$1\%$$
This may also be expressed as a confidence level of:
$$99\%$$
The \(0.01\) significance level is stricter than the \(0.05\) level.
A result must provide stronger evidence against the null hypothesis to be significant at:
$$p\leq0.01$$
than at:
$$p\leq0.05$$
The consequences of selecting different significance levels are developed further in Type I and Type II errors.
Comparing significance levels
Significance level | Maximum probability attributed to chance | Confidence level | Relative strictness |
\(p\leq0.05\) | \(5\%\) | \(95\%\) | Common threshold |
\(p\leq0.01\) | \(1\%\) | \(99\%\) | Stricter threshold |
If a result is statistically significant at:
$$p\leq0.01$$
it will also meet the less strict criterion of:
$$p\leq0.05$$
However, a result significant at:
$$p\leq0.05$$
will not necessarily be significant at:
$$p\leq0.01$$
Statistical significance
A result is statistically significant when it meets the selected probability criterion.
A statistically significant result suggests that the observed pattern is sufficiently unlikely to have occurred through chance variation alone.
When a result is statistically significant:
the null hypothesis is rejected;
the alternative hypothesis receives support;
the researcher concludes that there is sufficient statistical evidence for a difference, correlation or association.
A significant result does not provide absolute proof. Statistical conclusions are based on probability rather than certainty.
A non-significant result
A result is not statistically significant when it does not meet the selected probability criterion.
This may be expressed as:
$$p>0.05$$
when the selected significance level was:
$$p\leq0.05$$
The symbol:
$$>$$
means greater than.
When a result is not statistically significant:
the null hypothesis is retained;
there is insufficient evidence to support the alternative hypothesis;
the apparent result may be due to chance variation.
A non-significant result does not prove that the null hypothesis is true.
It means that the study did not produce sufficient evidence to reject it.
Significant and non-significant outcomes
Statistical outcome | Probability statement | Decision about the null hypothesis |
Significant at the \(0.05\) level | \(p\leq0.05\) | Reject the null hypothesis |
Significant at the \(0.01\) level | \(p\leq0.01\) | Reject the null hypothesis |
Not significant at the \(0.05\) level | \(p>0.05\) | Retain the null hypothesis |
What \(p\leq0.05\) does not mean
The statement:
$$p\leq0.05$$
does not mean that:
there is a \(5\%\) chance that the research hypothesis is false;
there is a \(95\%\) chance that the research hypothesis is true;
the result is definitely genuine;
the result will always be replicated;
the effect is large or important;
one variable caused another.
It means that the result met a probability criterion based on the assumption that the null hypothesis is true.
Statistical significance and effect size
Statistical significance does not tell the researcher how large or practically important a result is.
For example, a very small difference might be statistically significant if the evidence is sufficiently consistent.
A larger-looking difference might not be statistically significant if:
the sample is small;
the scores vary greatly;
the pattern is inconsistent.
The statistical conclusion must therefore be based on the inferential test rather than visual inspection of means, percentages or graphs.
Statistical significance and causation
A statistically significant correlation does not demonstrate cause and effect.
Suppose a researcher finds:
$$p\leq0.05$$
for a correlation between stress scores and sleep quality.
The researcher may conclude:
There is a statistically significant correlation between stress and sleep quality.
The researcher cannot conclude:
Stress definitely causes poor sleep quality.
The direction and strength of correlations are covered in scattergrams and correlation coefficients.
Directional hypotheses and one-tailed tests
A directional hypothesis predicts the direction of the result.
For example:
Participants will recall more words in silence than in background noise.
This predicts which condition will produce the higher score.
A directional hypothesis is tested using a one-tailed test.
When using a statistical table, the researcher selects the column for:
one-tailed testing;
the chosen significance level.
Non-directional hypotheses and two-tailed tests
A non-directional hypothesis predicts a difference or relationship but does not predict its direction.
For example:
There will be a difference in the number of words recalled in silence and in background noise.
This predicts a difference but does not say which condition will produce higher scores.
A non-directional hypothesis is tested using a two-tailed test.
When using a statistical table, the researcher selects the column for:
two-tailed testing;
the chosen significance level.
Comparing one-tailed and two-tailed tests
Feature | One-tailed test | Two-tailed test |
Type of alternative hypothesis | Directional | Non-directional |
Prediction | Specifies the direction | Predicts a result without specifying direction |
Table column | One-tailed | Two-tailed |
Example | Scores will be higher in Condition A | Scores will differ between Conditions A and B |
The decision about whether a test is one-tailed or two-tailed must be based on the hypothesis, not on the direction of the result obtained.
Statistical tables
A statistical table contains critical values for a particular inferential test.
Different tests have different statistical tables.
A critical value may depend on:
the inferential test;
the number of participants or scores;
the value of \(N\);
degrees of freedom;
the chosen significance level;
whether the test is one-tailed or two-tailed.
The table provided in an examination will contain the information needed for the question.
The observed value
The observed value is the test statistic calculated from the research data.
Examples include:
$$S_{\text{observed}}$$
for the sign test, or a calculated correlation coefficient such as:
$$r_s=0.72$$
The observed value comes from the results of the study.
The critical value
The critical value is the threshold obtained from the statistical table.
It is used to judge whether the observed value is sufficiently extreme for the result to be statistically significant.
A useful distinction is:
The observed value is calculated from the data. The critical value is found in the table.
Finding a critical value
Use the following method.
Identify the correct inferential test.
Identify whether the hypothesis is directional or non-directional.
Identify the selected significance level.
Find the required sample information, such as \(N\) or degrees of freedom.
Locate the correct row in the statistical table.
Locate the correct significance and tail column.
Read the critical value at the intersection.
Compare the critical value with the observed value.
The selection of the test itself is covered in choosing an inferential test.
Selecting the correct significance column
Suppose the statistical table contains columns labelled:
Significance level | One-tailed | Two-tailed |
\(p\leq0.05\) | Column A | Column B |
\(p\leq0.01\) | Column C | Column D |
A directional hypothesis tested at:
$$p\leq0.05$$
requires Column A.
A non-directional hypothesis tested at:
$$p\leq0.05$$
requires Column B.
A directional hypothesis tested at:
$$p\leq0.01$$
requires Column C.
A non-directional hypothesis tested at:
$$p\leq0.01$$
requires Column D.
Observed and critical values
The observed and critical values must be compared using the rule for the selected test.
For some tests, the result is significant when:
$$\text{Observed value}\geq\text{Critical value}$$
For other tests, the result is significant when:
$$\text{Observed value}\leq\text{Critical value}$$
Do not apply the same comparison rule to every test.
Tests requiring an observed value equal to or greater than the critical value
For the following tests, the result is significant when the observed value is equal to or greater than the critical value:
Spearman’s rho;
Pearson’s \(r\);
related \(t\)-test;
unrelated \(t\)-test;
Chi-squared test.
The decision rule is:
$$\text{Observed value}\geq\text{Critical value}$$
These tests are covered in:
Worked example: greater observed value needed
A researcher calculates an observed correlation coefficient of:
$$r_s=0.68$$
The relevant statistical table gives a critical value of:
$$r_{s,\text{critical}}=0.62$$
For this test, the observed value must be equal to or greater than the critical value:
$$0.68\geq0.62$$
The result is therefore statistically significant.
A suitable conclusion is:
The observed value of \(0.68\) was greater than the critical value of \(0.62\). The result was statistically significant at the selected level, so the null hypothesis was rejected.
Worked example: greater-value test is not significant
A researcher obtains:
$$t_{\text{observed}}=1.74$$
The relevant critical value is:
$$t_{\text{critical}}=2.10$$
The decision rule requires:
$$t_{\text{observed}}\geq t_{\text{critical}}$$
However:
$$1.74<2.10$$
The result is not statistically significant.
At the \(0.05\) level, this may be reported as:
$$p>0.05$$
The null hypothesis should be retained.
Tests requiring an observed value equal to or smaller than the critical value
For the following tests, the result is significant when the observed value is equal to or smaller than the critical value:
sign test;
Wilcoxon test;
Mann-Whitney test.
The decision rule is:
$$\text{Observed value}\leq\text{Critical value}$$
The latter two tests are covered in Wilcoxon and Mann-Whitney tests.
Worked example: smaller observed value needed
A sign test produces:
$$S_{\text{observed}}=1$$
The relevant critical value is:
$$S_{\text{critical}}=2$$
For the sign test:
$$S_{\text{observed}}\leq S_{\text{critical}}$$
Compare the values:
$$1\leq2$$
The result is statistically significant.
The null hypothesis should be rejected.
Worked example: smaller-value test is not significant
A Mann-Whitney test produces:
$$U_{\text{observed}}=18$$
The critical value is:
$$U_{\text{critical}}=12$$
The result would be significant if:
$$U_{\text{observed}}\leq U_{\text{critical}}$$
However:
$$18>12$$
The result is not statistically significant.
The null hypothesis should be retained.
Equality meets the significance criterion
An observed value exactly equal to the critical value meets the required threshold.
For a test using:
$$\text{Observed value}\geq\text{Critical value}$$
the following result is significant:
$$8\geq8$$
For a test using:
$$\text{Observed value}\leq\text{Critical value}$$
the following result is also significant:
$$3\leq3$$
The equality sign matters.
Summary of decision rules
Statistical test | Significant when |
Sign test | \(\text{Observed}\leq\text{Critical}\) |
Wilcoxon | \(\text{Observed}\leq\text{Critical}\) |
Mann-Whitney | \(\text{Observed}\leq\text{Critical}\) |
Spearman’s rho | \(\text{Observed}\geq\text{Critical}\) |
Pearson’s \(r\) | \(\text{Observed}\geq\text{Critical}\) |
Related \(t\)-test | \(\text{Observed}\geq\text{Critical}\) |
Unrelated \(t\)-test | \(\text{Observed}\geq\text{Critical}\) |
Chi-squared | \(\text{Observed}\geq\text{Critical}\) |
A useful memory rule
For the sign test, Wilcoxon and Mann-Whitney:
A smaller observed value provides stronger evidence against the null hypothesis.
For correlation coefficients, \(t\)-tests and Chi-squared:
A larger observed value provides stronger evidence against the null hypothesis.
Always check the rule for the test named in the question.
Example statistical-table extract
The following extract shows critical values for a sign test at:
$$p\leq0.05$$
\(N\) | One-tailed critical value | Two-tailed critical value |
\(10\) | \(1\) | \(1\) |
\(11\) | \(2\) | \(1\) |
\(12\) | \(2\) | \(2\) |
Suppose:
\(N=11\);
the hypothesis is directional;
the significance level is \(p\leq0.05\).
The researcher should use:
the row for \(N=11\);
the one-tailed column.
The critical value is:
$$2$$
If the observed value is:
$$S_{\text{observed}}=2$$
then:
$$2\leq2$$
The result is statistically significant.
Applying a stricter significance level
A critical value selected at:
$$p\leq0.01$$
sets a stricter threshold than one selected at:
$$p\leq0.05$$
This means that a more extreme observed value is normally required before the null hypothesis is rejected.
For a test where a larger value is needed, the critical value at \(0.01\) is normally more demanding than the value at \(0.05\).
For a test where a smaller value is needed, the observed value must normally be even smaller to reach the stricter threshold.
Stating a significant statistical conclusion
A complete significant conclusion should include:
The observed value.
The critical value.
The comparison between them.
The significance level.
The decision about the null hypothesis.
A conclusion referring to the study.
Example
A test produces:
$$\text{Observed value}=14.6$$
The critical value is:
$$\text{Critical value}=12.8$$
The test requires the observed value to be equal to or greater than the critical value.
The comparison is:
$$14.6\geq12.8$$
A complete conclusion would be:
The observed value of \(14.6\) was greater than the critical value of \(12.8\). The result was statistically significant at \(p\leq0.05\), so the null hypothesis was rejected. There was a statistically significant difference between the two experimental conditions.
Stating a non-significant statistical conclusion
Suppose:
$$\text{Observed value}=10.4$$
and:
$$\text{Critical value}=12.8$$
The test requires:
$$\text{Observed value}\geq\text{Critical value}$$
However:
$$10.4<12.8$$
A complete conclusion would be:
The observed value of \(10.4\) was lower than the critical value of \(12.8\). 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.
Appropriate statistical symbols
The following symbols may be used when reporting statistical conclusions.
Symbol | Meaning |
\(p\) | Probability |
\(\leq\) | Less than or equal to |
\(\geq\) | Greater than or equal to |
\(<\) | Less than |
\(>\) | Greater than |
\(=\) | Equal to |
\(p\leq0.05\) | Significant at the \(5\%\) level |
\(p\leq0.01\) | Significant at the \(1\%\) level |
\(p>0.05\) | Not significant at the \(5\%\) level |
Writing conventional probability statements
A significant result at the \(0.05\) level may be written as:
$$p\leq0.05$$
A significant result at the \(0.01\) level may be written as:
$$p\leq0.01$$
A result that is not significant at the \(0.05\) level may be written as:
$$p>0.05$$
Do not reverse the inequality.
The statement:
$$p\geq0.05$$
is not the conventional way to state that a result is significant at the \(0.05\) level.
Reporting a test statistic
A statistical conclusion may include both the test statistic and the significance level.
For example:
$$r_s=0.72,\ p\leq0.05$$
This states that:
the observed Spearman’s rho coefficient was \(0.72\);
the result was statistically significant at the \(0.05\) level.
A sign-test result might be written as:
$$S=1,\ N=11,\ p\leq0.05$$
The exact information included depends on the question and statistical test.
Statistical significance and the null hypothesis
When:
$$p\leq0.05$$
at a selected \(0.05\) level:
$$\text{Reject the null hypothesis}$$
When:
$$p>0.05$$
at a selected \(0.05\) level:
$$\text{Retain the null hypothesis}$$
Do not write that a non-significant result proves the null hypothesis.
Contextual conclusions
A conclusion should identify what was investigated.
For a test of difference:
There was a statistically significant difference in memory scores between the silence and background-noise conditions.
For a correlation:
There was a statistically significant positive correlation between hours of sleep and memory score.
For an association:
There was a statistically significant association between treatment condition and participant improvement.
Use:
difference for comparisons;
correlation for two co-variables;
association for categorical variables.
A complete statistical-testing process
Use the following sequence.
Step 1: Identify the purpose
Determine whether the study tests:
a difference;
a correlation;
an association.
Step 2: Identify the experimental design
For a test of difference, decide whether the data are:
related;
unrelated.
Step 3: Identify the level of measurement
Determine whether the data are:
nominal;
ordinal;
interval.
Step 4: Select the statistical test
Use the purpose, design and level of measurement.
Step 5: Identify the hypothesis direction
Determine whether the hypothesis is:
directional;
non-directional.
Step 6: Select the significance level
For example:
$$p\leq0.05$$
Step 7: Find the observed value
Calculate or identify the test statistic.
Step 8: Find the critical value
Use the correct:
statistical table;
sample information;
significance column;
tail column.
Step 9: Apply the correct comparison rule
Use either:
$$\text{Observed}\geq\text{Critical}$$
or:
$$\text{Observed}\leq\text{Critical}$$
depending on the test.
Step 10: State the conclusion
Include:
the probability statement;
whether the result is significant;
whether the null hypothesis is rejected or retained;
the meaning of the result in context.
Key Words 🔑
Key word | Student-friendly definition | How it may be used in an exam |
Probability | The likelihood that an event or result will occur. | You may explain the meaning of \(p\leq0.05\) or \(p\leq0.01\). |
\(p\) | The symbol used to represent probability. | It is used when reporting statistical significance. |
Significance level | The probability threshold selected for deciding whether a result is statistically significant. | You may use it to select the correct statistical-table column. |
Statistical significance | A judgement that a result meets the selected probability criterion. | You may determine significance by comparing observed and critical values. |
\(p\leq0.05\) | A result meeting a probability threshold of no more than \(5\%\). | It may be used to report a statistically significant result. |
\(p\leq0.01\) | A result meeting a probability threshold of no more than \(1\%\). | It represents a stricter significance criterion. |
\(p>0.05\) | A result that does not meet the \(0.05\) significance threshold. | It may be used when retaining the null hypothesis. |
Observed value | The test statistic calculated from the research results. | It is compared with the critical value. |
Critical value | The threshold found in a statistical table. | It determines whether the observed value is statistically significant. |
Statistical table | A table containing critical values for an inferential test. | You may select a value using \(N\), degrees of freedom, significance and tail. |
One-tailed test | A significance test used with a directional hypothesis. | It determines which statistical-table column should be used. |
Two-tailed test | A significance test used with a non-directional hypothesis. | It usually uses a different critical-value column. |
Directional hypothesis | A hypothesis predicting the direction of a difference or relationship. | It requires a one-tailed significance test. |
Non-directional hypothesis | A hypothesis predicting a difference or relationship without stating its direction. | It requires a two-tailed significance test. |
Null hypothesis | A statement that there is no genuine difference, correlation or association. | It is rejected when the result is significant. |
Alternative hypothesis | A statement predicting a difference, correlation or association. | It receives support when the null hypothesis is rejected. |
Degrees of freedom | A value used to select the relevant row in some statistical tables. | It may be needed when finding a critical value. |
Hints from the Examiner Reports 💡
No lesson-specific examiner guidance was identified in the provided reports.
Common Mistakes ⚠️
Mistake: Interpreting \(p\leq0.05\) as a \(5\%\) chance that the alternative hypothesis is wrong.
Why this is incorrect:
The probability statement concerns obtaining the result if the null hypothesis is true. It is not the probability that a hypothesis is true or false.
How to improve:
State that the result meets a threshold under which the probability attributed to chance is no more than \(5\%\).
Mistake: Saying that a significant result proves the alternative hypothesis.
Why this is incorrect:
Inferential testing is based on probability and does not provide absolute proof.
How to improve:
Write that the null hypothesis is rejected and the alternative hypothesis receives support.
Mistake: Saying that a non-significant result proves the null hypothesis.
Why this is incorrect:
A non-significant result means that there is insufficient evidence to reject the null hypothesis.
How to improve:
Use the phrase:
The null hypothesis is retained.
Mistake: Confusing the \(0.05\) and \(0.01\) levels.
Why this is incorrect:
The \(0.01\) level is stricter because it allows a smaller probability of obtaining the result through chance.
How to improve:
Remember:
$$0.01<0.05$$
Therefore, \(0.01\) is the stricter criterion.
Mistake: Choosing one-tailed or two-tailed significance from the results.
Why this is incorrect:
The choice depends on whether the alternative hypothesis is directional or non-directional.
How to improve:
Read the hypothesis before looking at the statistical table.
Mistake: Using the original sample size when the statistical table requires a different value.
Why this is incorrect:
Some tables require:
non-zero \(N\);
the number of pairs;
group sizes;
degrees of freedom.
How to improve:
Read the row heading and table instructions carefully.
Mistake: Confusing the observed and critical values.
Why this is incorrect:
The observed value is calculated from the data, while the critical value comes from the statistical table.
How to improve:
Remember:
Observed comes from observations. Critical comes from the critical-values table.
Mistake: Applying the same comparison rule to every test.
Why this is incorrect:
Some tests require:
$$\text{Observed}\geq\text{Critical}$$
Others require:
$$\text{Observed}\leq\text{Critical}$$
How to improve:
Learn the rule for the test named in the question.
Mistake: Treating equal observed and critical values as non-significant.
Why this is incorrect:
An observed value equal to the critical value has reached the threshold.
How to improve:
Include equality in the comparison:
$$\geq$$
or:
$$\leq$$
Mistake: Writing \(p\geq0.05\) for a result significant at the \(0.05\) level.
Why this is incorrect:
A significant result meets a probability criterion of less than or equal to the selected level.
How to improve:
Write:
$$p\leq0.05$$
Mistake: Giving only a symbolic conclusion.
Why this is incorrect:
A statement such as:
$$p\leq0.05$$
does not explain what the result means for the study.
How to improve:
Add a contextual conclusion naming the variables, conditions or groups.
Mistake: Assuming that statistical significance means practical importance.
Why this is incorrect:
A statistically significant effect may still be small or unimportant in everyday settings.
How to improve:
Distinguish the probability judgement from the size or usefulness of the result.
Exam-Style Questions ✍️
Question 1
Define probability in the context of inferential statistical testing.[2 marks]
Question 2
Explain what is meant by:
a) \(p\leq0.05\)[2 marks]
b) \(p\leq0.01\)[2 marks]
Question 3
Explain why the significance level:
$$p\leq0.01$$
is stricter than:
$$p\leq0.05$$
[2 marks]
Question 4
A researcher uses a directional hypothesis.
State whether a one-tailed or two-tailed critical value should be used. Explain your answer.[2 marks]
Question 5
A researcher calculates:
$$r_s=0.68$$
The critical value is:
$$0.62$$
For Spearman’s rho, the observed value must be equal to or greater than the critical value.
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 6
The following extract shows critical values for a sign test at:
$$p\leq0.05$$
\(N\) | One-tailed critical value | Two-tailed critical value |
\(10\) | \(1\) | \(1\) |
\(11\) | \(2\) | \(1\) |
\(12\) | \(2\) | \(2\) |
A researcher has:
\(N=11\);
a non-directional hypothesis;
an observed value of \(S=1\).
a) Identify the appropriate critical value.[1 mark]
b) Determine whether the result is statistically significant.[1 mark]
c) Explain the decision using the correct sign-test comparison rule.[2 marks]
Question 7
A related \(t\)-test produces:
$$t_{\text{observed}}=1.86$$
The critical value is:
$$t_{\text{critical}}=2.12$$
For a \(t\)-test, the observed value must be equal to or greater than the critical value.
Write an appropriate statistical conclusion using:
the observed value;
the critical value;
an appropriate probability statement;
the decision about the null hypothesis.
[4 marks]
Question 8
A student writes:
“The result was significant at \(p\leq0.05\), so there is a \(95\%\) probability that the alternative hypothesis is true.”
Explain why this interpretation is incorrect.[3 marks]
Question 9
For each result below, state whether it is statistically significant.
a) A test requires the observed value to be equal to or greater than the critical value:
$$\text{Observed}=14,\quad\text{Critical}=11$$
[1 mark]
b) A test requires the observed value to be equal to or greater than the critical value:
$$\text{Observed}=9,\quad\text{Critical}=11$$
[1 mark]
c) A test requires the observed value to be equal to or smaller than the critical value:
$$\text{Observed}=3,\quad\text{Critical}=3$$
[1 mark]
d) A test requires the observed value to be equal to or smaller than the critical value:
$$\text{Observed}=6,\quad\text{Critical}=3$$
[1 mark]
Question 10
A psychologist investigates whether a relaxation programme changes participants’ anxiety ratings.
The statistical test produces:
$$\text{Observed value}=2$$
The relevant critical value at:
$$p\leq0.05$$
is:
$$3$$
For this test, the observed value must be equal to or smaller than the critical value.
Write a complete conclusion for the study. Your answer should:
compare the observed and critical values;
use an appropriate probability symbol;
state whether the result is statistically significant;
state whether the null hypothesis should be rejected or retained;
refer to participants’ anxiety ratings.
[5 marks]
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