Statistics and Probability — Free Maths AI SL Practice Questions

A researcher studies the relationship between the number of hours a plant is exposed to sunlight each day, $x$, and the height of the plant after 30 days, $y$ cm. Data collected for six plants is shown below. x (hours): 1, 2, 3, 4, 5, 6 y (cm): 12, 18, 25, 30, 36, 40

A biologist studies the relationship between temperature ($T$, in °C) and chirping rate ($C$, in chirps per minute) of a species of cricket. She collects data for $n = 8$ temperatures and obtains the following summary statistics: $$\sum T = 160, \quad \sum C = 140, \quad \sum T^2 = 3600, \quad \sum C^2 = 2800, \quad \sum TC = 3100$$

A marine biologist studies the relationship between water temperature ($T$, in °C) and diving depth ($D$, in metres) for a species of seal. Data from $n = 12$ seals give the following summary statistics: $$\sum T = 96.0, \quad \sum D = 240, \quad \sum T^2 = 824.0, \quad \sum D^2 = 5400, \quad \sum TD = 2100$$

A student investigates the relationship between the age of a car, $x$ years, and its selling price, $y$ thousand dollars. Data for five cars are shown in the table below. Age $x$ (years) — 1 — 2 — 3 — 4 — 5 Price $y$ (thousand dollars) — 18 — 15 — 12 — 9 — 6 The regression line of $y$ on $x$ is $y = 21.0 - 3.00x$.

A biologist is studying the relationship between the temperature ($x$, in $^\circ\text{C}$) and the rate of oxygen consumption ($y$, in mL/g/h) of a species of beetle. Data for six beetles are recorded in the table below. $x$ — 10 — 15 — 20 — 25 — 30 — 35 $y$ — 1.2 — 1.9 — 2.5 — 3.0 — 3.8 — 4.3 The following formulae are provided: $$r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{n\sum x^2 - (\sum x)^2}\;\sqrt{n\sum y^2 - (\sum y)^2}}$$ $$b = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}, \qquad a = \bar{y} - b\bar{x}$$

A small company records the number of days each of its 8 employees was absent from work last month. The data set is: $$2,\ 5,\ 3,\ 8,\ 2,\ 4,\ 5,\ 3$$

The ages (in years) of eight people in a cycling club are: $$22,\ 25,\ 27,\ 30,\ 33,\ 35,\ 38,\ y$$

The following table shows the number of hours spent on social media per week by a group of $20$ students. Hours — $0-4$ — $5-9$ — $10-14$ — $15-19$ — $20-24$ Frequency — $3$ — $7$ — $6$ — $3$ — $1$

A gardener measures the heights (in cm) of 7 sunflower plants: $$12,\ 15,\ 11,\ 18,\ 15,\ 14,\ 20$$

The test scores (out of 100) for a class of 10 students are listed in ascending order: $$45,\ 52,\ 58,\ 61,\ 63,\ 67,\ 71,\ 74,\ 78,\ z$$

A bag contains $12$ red marbles and $8$ blue marbles. Two marbles are selected at random from the bag without replacement.

In a school, $60\%$ of students play football and $45\%$ play basketball. It is known that $25\%$ of students play both sports.

In a medical screening test for a certain disease, the following probabilities are known: - The probability that a randomly selected person has the disease is $0.02$ - If a person has the disease, the probability that the test is positive is $0.95$ - If a person does not have the disease, the probability that the test is positive is $0.04$

A fair six-sided die is rolled twice. The faces are numbered $1, 2, 3, 4, 5, 6$.

A test for a rare disease has the following probabilities: if a person has the disease, the test is positive with probability $0.98$; if a person does not have the disease, the test is positive with probability $0.03$. It is known that $1\%$ of the population has the disease.

A factory produces electronic components. It is known that $8\%$ of the components are defective. A quality control inspector selects a random sample of $12$ components. The number of defective components, $X$, follows a binomial distribution.

A factory produces electronic components. It is known that $5\%$ of components are defective. A quality control inspector selects a random sample of $12$ components. Let $X$ represent the number of defective components in the sample.

A quality control engineer at a pharmaceutical company monitors the production of vaccine vials. Historical data shows that $3\%$ of vials have a contamination defect. The company uses a two-stage screening process. - First stage: an automated optical inspection detects $90\%$ of contaminated vials but incorrectly flags $2\%$ of clean vials as contaminated. Flagged vials are rejected; unflagged vials pass to the second stage. - Second stage: a chemical test, performed only on vials that passed the first stage, correctly identifies $98\%$ of contaminated vials and correctly clears $99\%$ of clean vials. Vials identified as contaminated in this stage are rejected; all others are released.

A fair six-sided die is rolled $8$ times. Let $X$ be the number of times a score of $4$ or greater is obtained.

A fair six-sided die is rolled $10$ times. Let the random variable $X$ denote the number of times a score of $4$ or greater is obtained.

The owner of a small bakery records the number of hours of sunshine, $h$, and the daily sales of iced coffee, $s$ (in cups), over 8 randomly selected summer days. $h$ (hours) — $4.2$ — $5.1$ — $6.8$ — $7.3$ — $8.0$ — $9.5$ — $10.2$ — $11.4$ $s$ (cups) — $18$ — $22$ — $31$ — $35$ — $38$ — $47$ — $51$ — $58$

A coffee shop records the outside temperature ($T\,°\text{C}$) and the number of hot chocolates sold ($N$) each day for 10 days. The data are shown in the scatter below. The regression line of $N$ on $T$ is $N = 38.2 - 0.95T$.

A researcher studies the relationship between hours of study ($h$ hours) and exam score ($s$ marks) for 10 students. The data are shown in the table below. $h$ — 2 — 4 — 6 — 8 — 10 — 12 — 14 — 16 — 18 — 20 $s$ — 35 — 42 — 55 — 60 — 68 — 75 — 78 — 85 — 90 — 92

A biologist is studying the relationship between the temperature ($T$, in $^\circ\text{C}$) and the rate of chirping ($C$, in chirps per minute) of a species of cricket. She collects the following data from eight crickets: $T$ ($^\circ\text{C}$) — 15 — 18 — 20 — 22 — 25 — 27 — 30 — 32 $C$ (chirps/min) — 20 — 24 — 28 — 31 — 36 — 39 — 44 — 48

A farmer measures the amount of fertiliser ($F$, in kg per hectare) applied to six plots of land the corresponding yield of wheat ($Y$, in tonnes per hectare). The data are shown below. $F$ (kg/ha) — 50 — 80 — 100 — 120 — 150 — 180 $Y$ (t/ha) — 3.2 — 4.5 — 5.1 — 5.8 — 6.4 — 7.0

The weekly screen time (in hours) for a random sample of 12 students from a large international school is recorded below: $$18,\; 22,\; 15,\; 30,\; 25,\; 20,\; 22,\; 28,\; 16,\; 22,\; 35,\; 19$$ -

The box-and-whisker below shows the distribution of test scores (out of 100) for a group of 40 students in a mathematics class.

The monthly salaries, in dollars, of 11 employees at a small company are: $$2800,\ 3200,\ 3500,\ 3700,\ 4000,\ 4100,\ 4200,\ 4500,\ 4800,\ 5200,\ 9500$$

The ages of 9 people in a yoga class are recorded as: $$22,\ 25,\ 27,\ 29,\ 30,\ 31,\ 34,\ 37,\ 41$$

A small business records the number of orders received each day over a 10-day period: $$12,\ 15,\ 8,\ 12,\ 20,\ 12,\ 18,\ 15,\ 10,\ 14$$

A survey of 200 students at an international school asked about their participation in two extracurricular activities: Model United Nations $(M)$ and Debating $(D)$. The results showed that 85 students participate in Model United Nations, 60 students participate in Debating, and 35 students participate in both activities.

A bag contains $4$ red marbles, $6$ blue marbles, and $10$ green marbles. Two marbles are drawn from the bag without replacement.

A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn from the bag without replacement.

In a factory, $3\%$ of the electronic components produced are defective. A quality control test correctly identifies a defective component as defective $96\%$ of the time. The test also incorrectly identifies a non-defective component as defective $2\%$ of the time.

A fair six-sided die is rolled twice. Let $A$ be the event that the sum of the two rolls is $7$, and let $B$ be the event that least one roll shows a $5$.

A factory produces electronic components. The probability that a randomly selected component is defective is $0.04$. A quality control inspector selects a random sample of $15$ components. Let $X$ be the number of defective components in the sample.

A medical test for a rare disease has a $92\%$ chance of correctly identifying a person with the disease (true positive) and a $95\%$ chance of correctly identifying a person without the disease (true negative). The disease affects $2\%$ of the population.

A factory produces electronic components. It is known that $92\%$ of the components pass a quality control test. A random sample of $15$ components is selected and tested. Let $X$ be the number of components that pass the test.

A biologist is studying a species of plant. It is known that $75\%$ of seeds from this species germinate successfully. The biologist plants $20$ seeds and monitors their growth. Let $X$ be the number of seeds that germinate.

A medical research team is testing a new treatment for a disease. In a clinical trial, the probability that a patient shows improvement after receiving the treatment is $0.65$. The trial involves $20$ randomly selected patients. Let $X$ be the number of patients who show improvement, where $X \sim B(20,\, 0.65)$.