Inferential statistics (List B)

Introduction to Inferential Statistics

Welcome to one of the most important parts of psychological research! So far, you have learned how to describe data using averages (like the mean) or graphs. But how do we know if the results of an experiment are "real" or if they just happened by pure luck? This is where inferential statistics come in.

Inferential statistics allow psychologists to make "inferences" (logical conclusions) about a whole population based on a small sample of people. They help us decide whether to accept our alternative hypothesis or stick with our null hypothesis. Don't worry if this feels a bit like "maths class" at first—we will break it down step-by-step!

1. Key Concepts: Probability and Significance

In psychology, we never say we have "proved" something 100%. Instead, we talk about probability (represented by the letter \(p\)). We want to know: "What is the probability that these results happened by chance?"

Significance Levels

The standard level of significance used in psychology is \(p \leq .05\). This means there is a 5% (or less) probability that the results occurred by fluke, and a 95% probability that the effect was caused by the variables we were studying.

Sometimes, researchers use stricter or more lenient levels:
• \(p \leq .10\): 10% level (more lenient, used in pilot studies).
• \(p \leq .01\): 1% level (very strict, used when we need to be extra sure, like in medical research).

One-Tailed vs. Two-Tailed Testing

Which test you use depends on your hypothesis (which you learned about in Topic B):
• One-tailed test: Use this if you have a directional hypothesis (e.g., "Scores will increase...").
• Two-tailed test: Use this if you have a non-directional hypothesis (e.g., "There will be a difference...").

Quick Tip: If your hypothesis predicts a specific direction, use one tail. If it just says "there will be a change" without saying which way, use two tails!

2. The "List B" Inferential Tests

According to the Edexcel syllabus, there are three specific tests you need to know. You don't need to memorize the formulas (they are given in the exam), but you do need to know when to use them.

A. Wilcoxon Signed Ranks Test

This test is usually introduced in Topic B (Cognitive Psychology).
• Used for: Looking for a difference between two conditions.
• Experimental Design: Used with repeated measures (the same people do both tasks) or matched pairs.
• Data Type: Ordinal data (data that can be ranked/put in order).

B. Spearman’s Rank Correlation Coefficient

This test is usually introduced in Topic C (Biological Psychology).
• Used for: Looking for a relationship or correlation between two co-variables.
• Example: Is there a link between the amount of a hormone in the blood and a person's aggression score?
• Data Type: Ordinal data.
• The Formula: \(\rho = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}\)

C. Chi-Squared (\(\chi^2\))

This test is usually introduced in Topic D (Learning Theories).
• Used for: Looking for a difference or an association.
• Data Type: Nominal data (data in categories, like "Yes/No", "Pass/Fail", or "Red/Blue").
• The Formula: \(\chi^2 = \sum \frac{(o - e)^2}{e}\)
• Degrees of Freedom (df): For Chi-Squared, you also need to calculate \(df = (r - 1) \times (c - 1)\) (rows minus one times columns minus one).

3. How to Choose the Right Test

In the exam, you might be asked to justify why a researcher chose a specific test. Use this simple checklist:
1. Is it a difference or a relationship? (Relationship = Spearman's).
2. What is the design? (Repeated measures = Wilcoxon).
3. What is the level of data? (Categories/Nominal = Chi-Squared).

Key Takeaway:
• Spearman's = Relationship + Ordinal.
• Wilcoxon = Difference + Repeated Measures + Ordinal.
• Chi-Squared = Difference/Association + Nominal.

4. Using Critical Value Tables

Once a researcher calculates a test statistic (the observed value), they must compare it to a critical value found in a statistical table. This tells them if their result is significant.

How to find the critical value:

1. Identify the significance level (usually \(p \leq .05\)).
2. Identify if the hypothesis is one-tailed or two-tailed.
3. Identify the N value (number of participants) or df (degrees of freedom).

The Decision Rule:

For the tests in List B, the rule is as follows:
• Spearman’s Rank: Observed value must be equal to or greater than the critical value.
• Chi-Squared: Observed value must be equal to or greater than the critical value.
• Wilcoxon: Observed value must be equal to or less than the critical value. (Careful! This one is the "odd one out.")

Summary: If the result is significant, we reject the null hypothesis and accept the alternative hypothesis. If it's not significant, we accept the null hypothesis.

5. Type I and Type II Errors

Even with perfect statistics, we can still make mistakes because we are dealing with probabilities.

Type I Error ("The False Positive")

This happens when we reject the null hypothesis when we should have accepted it. We think we found a significant effect, but actually, the results were just down to chance.
Analogy: A doctor telling a man he is pregnant. It's a false alarm!
• Risk: Usually happens if our significance level is too lenient (e.g., \(p \leq .10\)).

Type II Error ("The False Negative")

This happens when we accept the null hypothesis when we should have rejected it. There was a real effect there, but we missed it!
Analogy: A doctor telling a heavily pregnant woman she isn't pregnant. A missed discovery!
• Risk: Usually happens if our significance level is too strict (e.g., \(p \leq .01\)) or the sample size is too small.

Quick Review:
• Type I = I erroneously said it's significant.
• Type II = I I-gnored a real effect (False Negative).

6. Summary Table

Test Name | Purpose | Data Level | Design
Spearman's | Correlation | Ordinal | N/A (Co-variables)
Wilcoxon | Difference | Ordinal | Repeated Measures / Matched Pairs
Chi-Squared | Difference | Nominal | Independent Groups

Key Takeaway: Inferential statistics are the "judge" of psychological research. They tell us if our hard work in the lab has actually revealed something meaningful about human behaviour or if we need to go back to the drawing board!