4.8.8Additional lessons
Statistical Analysis of Dihybrid Crosses — IB Biology HL Revision Notes & Topical Questions
5 min read 5 parts Lesson 75 of 101
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Section 1 of 5The Idea Behind the Test
Why test
- Why test Real offspring numbers rarely match a ratio exactly, so a statistical test decides whether the difference is due to chance alone.
Null hypothesis
- Null hypothesis There is no significant difference between the observed and expected results, for example that the offspring fit a 9 : 3 : 3 : 1 ratio.
Type of data
- Type of data The chi-squared test is used for counts in categories, such as numbers of each phenotype, never for measurements.
Sample size
- Sample size The expected count in each category should be at least 5, otherwise the test becomes unreliable.
Section 2 of 5Carrying Out Chi-Squared
The formula compares each observed value O with its expected value E.
1
State the null hypothesis and the expected ratio, then calculate the expected number in each class by multiplying the total by its fraction of the ratio.
2
Calculate (O − E)² ÷ E for every class, squaring the difference so negative values do not cancel.
3
Add the values together: χ² = Σ (O − E)² ÷ E.
4
Work out the degrees of freedom as the number of classes minus one, so four phenotype classes give 3 degrees of freedom.
5
Compare the calculated χ² with the critical value at the 0.05 probability level to decide whether to reject the null hypothesis.
| Degrees of freedom | Critical value at p = 0.05 |
|---|---|
| 1 | 3.84 |
| 2 | 5.99 |
| 3 | 7.82 |
Section 3 of 5Interpreting the Result
Interpreting the Result
Calculated value smaller
- Calculated value smaller If χ² is below the critical value, the difference is not significant and chance can explain it,
- so the null hypothesis is accepted.
Calculated value larger
- Calculated value larger If χ² is above the critical value, the difference is significant at the 5% level, so the null hypothesis is rejected.
Meaning of p = 0.05
- Meaning of p = 0.05 There is a 5% probability that a difference this large would arise by chance if the null hypothesis were true.
Possible causes
- Possible causes A significant result suggests another factor, such as gene linkage or lower viability of some genotypes.
Section 4 of 5Worked Example
Question
- Two heterozygous plants are crossed and 160 offspring are counted: 94 round yellow, 28 round green, 31 wrinkled yellow and 7 wrinkled green.
- Test the fit to a 9 : 3 : 3 : 1 ratio.
- Expected counts9/16 × 160 = 90, 3/16 × 160 = 30, 30, and 1/16 × 160 = 10.
- Calculate (O − E)² ÷ E
- (94 − 90)²/90 = 0.178;
- (28 − 30)²/30 = 0.133;
- (31 − 30)²/30 = 0.033;
- (7 − 10)²/10 = 0.900.
- (94 − 90)²/90 = 0.178;
- Addχ² = 0.178 + 0.133 + 0.033 + 0.900 = 1.244 (about 1.24).
- Degrees of freedom = 4 − 1 = 3, so the critical value at p = 0.05 is 7.82.
- 1.24 is less than 7.82.
Answer The difference is not significant at the 0.05 level, so the null hypothesis is accepted: the results fit a 9 : 3 : 3 : 1 ratio.
Section 5 of 5Exam Checklist & Tips
4.8.8 – Key Points for the Exam
- Null hypothesis The statement that there is no difference between observed and expected results.
- Chi-squared test A test comparing observed and expected counts in categories.
- Degrees of freedom The number of categories minus one.
- Critical value The χ² value at a chosen probability level above which the difference is significant.
Common Mistakes
- Using percentages instead of actual counts in the chi-squared formula.
- Calculating degrees of freedom as the number of classes instead of classes minus one.
- Saying the null hypothesis is 'proved' when χ² is small; it is only accepted for this data.
- Forgetting to square (O − E), or dividing by O instead of E.
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