Statistics and probability practice questions (IB Maths AI)

Sample questions

All 4 free IB Maths AI questions on this topic, from the free IB Maths AI SL Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 16 marks · Descriptive statistics
   The times, in minutes, taken by 12 students to complete a puzzle were 14, 18, 22, 23, 25, 25, 27, 29, 31, 34, 38 and 52.
   (a)

   Write down the median time.

   [1]
   (b)

   Find the interquartile range.

   [2]
   (c)

   Show that the time of 52 minutes is an outlier.

   [2]
   (d)

   Find the mean time.

   [1]
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2. Question 26 marks · Normal & binomial distributions
   The masses of the apples grown on a farm are normally distributed with mean 165 g and standard deviation 12 g.
   (a)

   Find the probability that a randomly chosen apple has a mass greater than 180 g.

   [2]
   (b)

   The heaviest 5% of the apples are sold as premium apples. Find the minimum mass of a premium apple.

   [2]
   (c)

   Six apples are chosen at random. Find the probability that at least one of them has a mass greater than 180 g.

   [2]
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3. Question 36 marks · Conditional probability
   A factory makes light bulbs on two machines. Machine A makes 60% of the bulbs, and 3% of its bulbs are faulty. Machine B makes the remaining 40% of the bulbs, and 5% of its bulbs are faulty. A bulb is chosen at random from the factory's output.
   (a)

   Find the probability that the bulb is faulty.

   [2]
   (b)

   Given that the bulb is faulty, find the probability that it was made by machine B.

   [2]
   (c)

   Two bulbs are chosen at random and independently. Find the probability that exactly one of them is faulty.

   [2]
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4. Question 410 marks · Correlation, regression & chi-square test
   A teacher records the number of hours, $x$, that each of eight students spent revising for a test, and their scores, $y$, out of 100. The pairs $(x, y)$ are (1, 46), (2, 50), (3, 58), (4, 57), (5, 66), (6, 70), (7, 71) and (8, 82). The teacher then surveys 150 students across the school about their main revision method and whether they passed the test. Of the 50 students who mainly re-read their notes, 28 passed; of the 40 who used flashcards, 26 passed; of the 60 who did practice tests, 48 passed.
   (a)

   Find Pearson's product-moment correlation coefficient, $r$, and describe the correlation.

   [2]
   (b)

   Write down the equation of the regression line of $y$ on $x$.

   [2]
   (c)

   Use your regression line to estimate the score of a student who revised for 5.5 hours.

   [1]
   (d)

   The teacher carries out a chi-square test of independence at the 5% significance level. Write down the null hypothesis.

   [1]
   (e)

   Show that the expected number of students who did practice tests and passed is 40.8.

   [1]
   (f)

   Find the chi-square statistic and the p-value for the test.

   [2]
   (g)

   State, with a reason, the conclusion of the test.

   [1]
   Show answerFrom the free IB Maths AI SL Practice Exam →

Key terms

The definitions and formulas IB Maths AI expects for statistics and probability, typeset as they appear on the paper.

Measures of central tendency and spread

Mean, median and mode; range, interquartile range, variance and standard deviation. For grouped data use mid-interval values and say so; the GDC gives all of them from a frequency table.

$$IQR = Q_3 - Q_1, \qquad \text{outlier if } x < Q_1 - 1.5\,IQR \text{ or } x > Q_3 + 1.5\,IQR$$

Pearson's and Spearman's correlation coefficients

Pearson's $r$ measures linear correlation on the raw data; Spearman's $r_s$ is Pearson's $r$ on the ranks and is less affected by outliers and suited to monotonic but non-linear relationships. Both lie between $-1$ and $1$.

Regression line of y on x

The least-squares line used to predict $y$ from $x$; it passes through the mean point $(\bar{x}, \bar{y})$. Predicting outside the data range (extrapolation) is unreliable and examiners expect you to say so.

$$y = ax + b$$

Chi-squared test for independence

Tests whether two categorical variables are independent using a contingency table. Compare the $p$-value with the significance level, or the statistic with the critical value; expected frequencies come from row and column totals.

$$\chi^2 = \sum \frac{(f_o - f_e)^2}{f_e}, \qquad \text{df} = (r - 1)(c - 1)$$

Chi-squared goodness of fit test

Tests whether observed frequencies match a given distribution (uniform, binomial, normal). Degrees of freedom are the number of categories minus one, minus one more for each parameter estimated from the data.

Two-sample t-test

Tests whether two population means differ, assuming normal populations with equal variance (pooled). State the hypotheses, read the $p$-value from the GDC and conclude in context.

$$H_0: \mu_1 = \mu_2, \qquad H_1: \mu_1 \ne \mu_2 \text{ (or } < , > \text{)}$$

Binomial and normal distributions

The binomial counts successes in $n$ independent trials with probability $p$; the normal is the continuous bell curve with mean $\mu$ and standard deviation $\sigma$. Both are evaluated on the GDC once the distribution is stated.

$$X \sim B(n, p), \quad E(X) = np; \qquad Y \sim N(\mu, \sigma^2), \quad z = \frac{y - \mu}{\sigma}$$

Conditional probability

The probability of $A$ given $B$, read from a Venn diagram, tree diagram or two-way table by restricting to the cases where $B$ happened.

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

Frequently asked questions

Other IB Maths AI topics

• Number and algebra
• Functions
• Geometry and trigonometry
• Calculus

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