A Level Biology - How to Solve: Student’s t-Test & Chi-Squared Test (Null Hypothesis, p-Values)

How to Solve: Student’s t-Test & Chi-Squared Test (Null Hypothesis, p-Values)

Complete Guide for GCSE/A-Level (Physics, Chemistry, Biology)

Introduction

"Mastering t-tests and chi-squared tests lets you prove whether a new drug works, if a physics experiment’s results are real, or if a genetic trait is truly linked—worth up to 10% of your A-Level Biology or Physics exam. Miss this, and you lose easy marks on data analysis questions."

WHAT YOU NEED TO KNOW FIRST

1. Null hypothesis (H₀): The default assumption that there’s no effect or no difference.
2. p-value: The probability of getting your results if the null hypothesis is true.
3. Degrees of freedom (df): A number that adjusts your test based on sample size.

KEY TERMS & FORMULAS

1. Student’s t-Test (Comparing Means)

When to use: Compare the means of two groups to see if they’re significantly different. Types: - Unpaired t-test: Two separate groups (e.g., drug vs. placebo). - Paired t-test: Same group before/after (e.g., blood pressure before/after treatment).

Formula (Unpaired t-test):

[ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} ] - (\bar{X}_1, \bar{X}_2) = means of group 1 and 2 (MEMORISE) - (s_1^2, s_2^2) = variances of group 1 and 2 (MEMORISE) - (n_1, n_2) = sample sizes (MEMORISE) - Degrees of freedom (df): (n_1 + n_2 - 2) (MEMORISE)

Critical t-value:

• Given in a table on the exam sheet.
• Compare your calculated (t) to the critical value at (p = 0.05) (5% significance).

2. Chi-Squared (χ²) Test (Categorical Data)

When to use: Check if observed results match expected results (e.g., genetic crosses, survey responses).

Formula:

[ \chi^2 = \sum \frac{(O - E)^2}{E} ] - (O) = observed frequency (MEMORISE) - (E) = expected frequency (MEMORISE) - Degrees of freedom (df): Number of categories - 1 (MEMORISE)

Critical χ² value:

• Given in a table on the exam sheet.
• Compare your calculated χ² to the critical value at (p = 0.05).

Key Rules for Both Tests:

1. Null hypothesis (H₀): "No difference" or "No effect."
2. Alternative hypothesis (H₁): "There is a difference" or "There is an effect."
3. p-value < 0.05: Reject H₀ (results are significant).
4. p-value ≥ 0.05: Fail to reject H₀ (not significant).

STEP-BY-STEP METHOD

For t-Test (Unpaired):

1. State hypotheses:
2. H₀: "No difference between the means."
3. H₁: "There is a difference between the means."
4. Calculate means ((\bar{X}_1, \bar{X}_2)) and variances ((s_1^2, s_2^2)).
5. Plug into the t-test formula.
6. Find degrees of freedom (df = (n_1 + n_2 - 2)).
7. Look up critical t-value at (p = 0.05) (given in table).
8. Compare calculated t to critical t:
9. If (|t| >) critical value → reject H₀ (significant difference).
10. If (|t| ≤) critical value → fail to reject H₀ (no significant difference).

For Chi-Squared (χ²) Test:

1. State hypotheses:
2. H₀: "No difference between observed and expected results."
3. H₁: "There is a difference between observed and expected results."
4. Calculate expected frequencies (E).
5. Plug observed (O) and expected (E) into χ² formula.
6. Find degrees of freedom (df = number of categories - 1).
7. Look up critical χ² value at (p = 0.05) (given in table).
8. Compare calculated χ² to critical χ²:
9. If χ² > critical value → reject H₀ (significant difference).
10. If χ² ≤ critical value → fail to reject H₀ (no significant difference).

WORKED EXAMPLES

Example 1 – Basic t-Test (Unpaired)

Question: A drug trial tests reaction times (ms) in two groups: - Group A (Drug): 120, 130, 140, 150, 160 - Group B (Placebo): 150, 160, 170, 180, 190 Is there a significant difference at (p = 0.05)?

Solution:
1. Hypotheses: - H₀: No difference in reaction times. - H₁: There is a difference in reaction times.
2. Calculate means: - (\bar{X}_A = \frac{120+130+140+150+160}{5} = 140) - (\bar{X}_B = \frac{150+160+170+180+190}{5} = 170)
3. Calculate variances: - (s_A^2 = \frac{(120-140)^2 + (130-140)^2 + ...}{5-1} = 250) - (s_B^2 = \frac{(150-170)^2 + (160-170)^2 + ...}{5-1} = 250)
4. Plug into t-test formula: [ t = \frac{140 - 170}{\sqrt{\frac{250}{5} + \frac{250}{5}}} = \frac{-30}{\sqrt{50 + 50}} = \frac{-30}{10} = -3 ]
5. Degrees of freedom: (5 + 5 - 2 = 8)
6. Critical t-value (df=8, p=0.05): 2.306 (from table)
7. Compare: (|-3| > 2.306) → Reject H₀ (significant difference).

What we did and why: - We compared two groups’ means using a t-test. - The calculated t-value was larger than the critical value, so we rejected the null hypothesis.

Example 2 – Medium Chi-Squared Test

Question: A genetics experiment crosses two heterozygous pea plants (Yy × Yy). Expected ratio: 3 yellow : 1 green. Observed: 72 yellow, 18 green. Is this significantly different from expected?

Solution:
1. Hypotheses: - H₀: Observed matches expected (3:1 ratio). - H₁: Observed does not match expected.
2. Calculate expected frequencies: - Total = 72 + 18 = 90 - Expected yellow = (90 \times \frac{3}{4} = 67.5) - Expected green = (90 \times \frac{1}{4} = 22.5)
3. Plug into χ² formula: [ \chi^2 = \frac{(72 - 67.5)^2}{67.5} + \frac{(18 - 22.5)^2}{22.5} = \frac{20.25}{67.5} + \frac{20.25}{22.5} = 0.3 + 0.9 = 1.2 ]
4. Degrees of freedom: 2 categories - 1 = 1
5. Critical χ² (df=1, p=0.05): 3.841 (from table)
6. Compare: 1.2 < 3.841 → Fail to reject H₀ (no significant difference).

What we did and why: - We checked if observed results matched the expected 3:1 ratio. - The χ² value was smaller than the critical value, so we kept the null hypothesis.

Example 3 – Exam-Style (Disguised t-Test)

Question: A student measures the mass of two types of nanoparticles: - Type X: 5.1, 5.3, 5.2, 5.4, 5.0 mg - Type Y: 4.8, 4.9, 5.0, 4.7, 4.6 mg The student claims Type X is heavier. Test this at (p = 0.05).

Solution:
1. Hypotheses: - H₀: No difference in mass. - H₁: Type X is heavier (one-tailed test).
2. Calculate means: - (\bar{X} = 5.2), (\bar{Y} = 4.8)
3. Calculate variances: - (s_X^2 = 0.025), (s_Y^2 = 0.025)
4. Plug into t-test formula: [ t = \frac{5.2 - 4.8}{\sqrt{\frac{0.025}{5} + \frac{0.025}{5}}} = \frac{0.4}{0.1} = 4 ]
5. Degrees of freedom: (5 + 5 - 2 = 8)
6. Critical t-value (df=8, p=0.05, one-tailed): 1.860 (from table)
7. Compare: 4 > 1.860 → Reject H₀ (Type X is significantly heavier).

What we did and why: - We used a one-tailed t-test because the student claimed a direction (heavier). - The t-value was larger than the critical value, so we supported the claim.

COMMON MISTAKES

1. MISTAKE: Using the wrong test (t-test for categorical data or χ² for means). WHY IT HAPPENS: Confusing when to use each test. CORRECT APPROACH: t-test for means, χ² for categories.

2. MISTAKE: Forgetting to square the numerator in χ². WHY IT HAPPENS: Misremembering the formula. CORRECT APPROACH: Always use ((O - E)^2).

3. MISTAKE: Using the wrong degrees of freedom. WHY IT HAPPENS: Not memorising df rules. CORRECT APPROACH: t-test: (n_1 + n_2 - 2), χ²: categories - 1.

4. MISTAKE: Ignoring one-tailed vs. two-tailed tests. WHY IT HAPPENS: Not reading the question carefully. CORRECT APPROACH: If the question specifies a direction (e.g., "heavier"), use one-tailed.

5. MISTAKE: Misinterpreting p-values. WHY IT HAPPENS: Confusing "significant" with "important." CORRECT APPROACH: p < 0.05 means reject H₀, not that the effect is large.

EXAM TRAPS

1. TRAP: The question gives raw data but expects you to calculate means/variances. HOW TO SPOT IT: If data is given as a list, you’ll need to process it first. HOW TO AVOID IT: Always check if you need to calculate means/variances before plugging into the formula.

2. TRAP: The question asks for a one-tailed test but doesn’t say so explicitly. HOW TO SPOT IT: Look for words like "greater," "less," "heavier," or "faster." HOW TO AVOID IT: If the question implies a direction, use a one-tailed test.

3. TRAP: The critical value table is for two-tailed tests, but you need one-tailed. HOW TO SPOT IT: The question specifies a direction. HOW TO AVOID IT: Divide the p-value by 2 (e.g., use 0.025 instead of 0.05 for one-tailed).

1-MINUTE RECAP

"Here’s what you need to remember for t-tests and chi-squared tests:
1. t-test: Compare means of two groups. Use the formula, find df, compare to critical t.
2. Chi-squared: Compare observed vs. expected categories. Use the formula, find df, compare to critical χ².
3. Null hypothesis (H₀): Always assume no difference first.
4. p-value < 0.05? Reject H₀—your results are significant.
5. Degrees of freedom: t-test = (n_1 + n_2 - 2), χ² = categories - 1.
6. One-tailed vs. two-tailed: If the question implies a direction, use one-tailed. Now go ace that exam!"