IB Maths AI statistics and the GDC: regression, Spearman, chi-squared and the normal distribution

Which statistical tool answers which question

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Worked example 1: regression line and Pearson's r

Data: x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5. Find the regression line of y on x and r.

1. Enter x and y as two lists on the GDC and run linear regression. It returns y = 0.6x + 2.2 and r = 0.775 (3 s.f.).
2. Check by hand: the means are 3 and 4. The sum of (x - 3)(y - 4) is 4 + 0 + 0 + 0 + 2 = 6, and the sum of (x - 3)^2 is 10, so the gradient is 6/10 = 0.6 and the intercept is 4 - 0.6 × 3 = 2.2.
3. r = 6/sqrt(10 × 6) = 6/sqrt(60) = 0.7746, which rounds to 0.775. The sum of (y - 4)^2 is 4 + 0 + 1 + 0 + 1 = 6.
4. Interpret: a fairly strong positive linear correlation. Predict y when x = 3.5: 0.6 × 3.5 + 2.2 = 4.3. This is interpolation, inside the data range, so it is reasonable.
5. Do not predict for x = 20. That is extrapolation, and examiners expect you to say the prediction is unreliable.

Worked example 2: Spearman's rank

Five students score 56, 72, 64, 88, 80 in maths and 60, 70, 75, 85, 78 in physics. Find r_s.

1. Rank each list from lowest to highest. Maths ranks: 1, 3, 2, 5, 4. Physics ranks: 1, 2, 3, 5, 4.
2. On the GDC, enter the ranks as two lists and find r for the ranks. That value is r_s = 0.9.
3. Check with the formula: the rank differences d are 0, 1, -1, 0, 0, so the sum of d^2 is 2. r_s = 1 - 6 × 2/(5 × (25 - 1)) = 1 - 12/120 = 0.9.
4. Interpret: a strong positive agreement between the rankings. Spearman's rank detects any consistently increasing relationship, not only a straight line, which is why it suits ranked or curved data.

Worked example 3: chi-squared test for independence

100 students are asked whether they prefer studying in the morning or the evening. In group X, 30 prefer morning and 20 prefer evening. In group Y, 20 prefer morning and 30 prefer evening. Test at the 5% level whether preference is independent of group.

1. Hypotheses. H0: study preference is independent of group. H1: study preference is not independent of group.
2. Expected frequencies: each row total is 50 and each column total is 50, so each expected value is 50 × 50/100 = 25. All expected values are at least 5, so the test is valid.
3. Degrees of freedom: (rows - 1)(columns - 1) = 1 × 1 = 1.
4. Statistic: each cell gives (O - E)^2/E = 25/25 = 1, so chi-squared = 4. Entering the observed table in the GDC gives chi-squared = 4 and p = 0.0455.
5. Conclusion: p = 0.0455 is less than 0.05, so reject H0. There is evidence at the 5% level that study preference depends on group. Using the critical value instead: 4 > 3.841 gives the same decision.

Worked example 4: the normal distribution

Test marks are modelled by X ~ N(65, 10^2).

1. P(X > 80): use the normal cumulative function with lower bound 80, a very large upper bound, mean 65 and standard deviation 10. Answer 0.0668 (3 s.f.). Check: z = (80 - 65)/10 = 1.5, and P(Z > 1.5) = 0.0668.
2. The mark needed to be in the top 10%: use the inverse normal with area 0.9 to the left. The GDC gives 77.8. Check: z = 1.2816, and 65 + 1.2816 × 10 = 77.8.
3. Always write the distribution and the probability statement, for example P(X > 80) = 0.0668. A bare number with no working risks losing the method mark if the answer is wrong.

Common mistakes

• Confusing standard deviation and variance. N(65, 10^2) means the standard deviation is 10; entering 100 into the calculator gives a wrong answer.
• Writing 'correlation proves causation'. A strong r shows association only.
• Stating H0 as 'the variables are dependent'. H0 is always independence.
• Comparing the chi-squared statistic with the p-value, or the p-value with the critical value. Compare like with like.
• Rounding early. Keep full calculator values until the final answer, then give 3 significant figures.
• Using the regression line of y on x to predict x from y.

Exam technique, and how one-to-one lessons help

Most AI statistics questions reward the same shape of answer: name the test or model, show the calculator inputs (lists, hypotheses, degrees of freedom), give the output to the right accuracy, then interpret it in one sentence about the real context. The interpretation mark is the one students most often drop, especially in Paper 2.

A tutor who teaches AI works with the student's own calculator model on screen, so every routine is practised exactly as it will be used in the exam. Lessons pair a short concept explanation on the shared whiteboard with past-paper questions where the student talks through each choice, and the tutor keeps a list of recurring slips, such as variance entered as standard deviation, until they stop happening.

Self-check

• For x = 2, 4, 6, 8 and y = 3, 7, 9, 13, find the regression line of y on x. (Answer: y = 1.6x, the intercept is 0)
• Ranks 1, 2, 3, 4 and 2, 1, 4, 3: find r_s. (Answer: 0.6)
• A 3 by 2 contingency table: how many degrees of freedom? (Answer: 2)
• X ~ N(50, 4^2): find P(X < 46). (Answer: 0.159)
• Explain in one sentence why predicting far outside the data range is unreliable.