IB Maths AA SL – Topic 4: Statistics and Probability
Descriptive statistics, probability, binomial and normal distributions, regression.
This worksheet covers:
- Descriptive statistics
- Probability (Venn diagrams, tree diagrams, conditional probability)
- The binomial
- Normal distributions
- Linear regression
A GDC is allowed throughout; allow about 75 minutes. Give numerical answers to 3 significant figures unless exact.
Questions
The number of goals scored by a hockey team in eight matches was$12,\ 15,\ 9,\ 18,\ 21,\ 15,\ 11,\ 24$.(a) Find the mean and the median.(b) Write down the mode and the range.(c) Find the standard deviation.
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The times, $t$ minutes, taken by 40 students to complete a puzzle are grouped as follows: $0 \le t < 10$: 4 students; $10 \le t < 20$: 9; $20 \le t < 30$: 15; $30 \le t < 40$: 8; $40 \le t < 50$: 4.(a) Write down the modal class.(b) Estimate the mean time.(c) Explain why your answer to (b) is only an estimate.
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A box-and-whisker diagram for a data set has minimum $5$, lower quartile $14$, median $19$, upper quartile $26$ and maximum $40$.(a) Find the interquartile range.(b) Determine, with a calculation, whether the value $40$ is an outlier.(c) The maximum is later corrected to $45$. State whether this changes your conclusion in (b).
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Events $A$ and $B$ are such that $\text{P}(A) = 0.4$, $\text{P}(B) = 0.5$ and $\text{P}(A \cap B) = 0.15$.(a) Find $\text{P}(A \cup B)$.(b) Find $\text{P}(A \mid B)$.(c) Determine whether $A$ and $B$ are independent. Justify your answer.
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A discrete random variable $X$ has probability distribution $\text{P}(X = x) = kx$ for $x = 1, 2, 3, 4$.(a) Show that $k = \dfrac{1}{10}$.(b) Find $\text{E}(X)$.(c) Find $\text{P}(X \ge 2 \mid X \le 3)$.
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A bag contains 5 red and 3 blue counters. Two counters are taken at random, one after the other, without replacement.(a) Find the probability that both counters are the same colour.(b) Find the probability that the second counter is red, given that the first counter is blue.(c) Find the probability that the first counter is blue, given that the second counter is red.
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A multiple-choice test has 10 questions, each with a probability $0.3$ of being answered correctly by a student who guesses. Let $X$ be the number of correct answers.(a) State the distribution of $X$ and any assumptions required.(b) Find $\text{P}(X = 3)$.(c) Find $\text{P}(X \ge 4)$.(d) Find the expected value and the variance of $X$.
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The heights of adult males in a town are normally distributed with mean $170$ cm and standard deviation $8$ cm.(a) Find the probability that a randomly chosen man is shorter than $160$ cm.(b) Find the probability that a randomly chosen man is between $165$ cm and $180$ cm tall.(c) The tallest $10$ percent of men are classed as ‘very tall’. Find the minimum height of a very tall man.
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The table shows the number of hours, $x$, six students spent on social media per week and their score, $y$, in a test.$x$: $2, 4, 5, 7, 8, 10$; $y$: $65, 58, 55, 48, 44, 37$.(a) Find Pearson’s product-moment correlation coefficient $r$ and describe the correlation.(b) Find the equation of the regression line of $y$ on $x$.(c) Use your line to estimate the score of a student who spends 6 hours per week on social media, and comment on the reliability of this estimate.(d) Explain why it would not be appropriate to use the line to estimate $y$ when $x = 30$.
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The mass of a bag of rice is normally distributed with mean $\mu$ g and standard deviation $5$ g. It is known that $20$ percent of bags have mass less than $50$ g.(a) Find $\mu$.(b) Bags with mass less than $50$ g are rejected. Eight bags are chosen at random. Find the probability that at least one bag is rejected.
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The time taken by a courier to deliver a parcel is normally distributed with mean $68$ minutes and standard deviation $6$ minutes. A delivery is ‘late’ if it takes more than $75$ minutes.(a) Show that the probability that a delivery is late is $0.122$, correct to 3 significant figures.(b) On one day the courier makes 8 deliveries. Find the probability that exactly 2 are late.(c) Find the probability that at least one delivery is late.(d) The courier’s manager wants the probability of a delivery being late to be $0.05$. Assuming the mean stays at $68$ minutes, find the standard deviation required.
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A scientist records the length, $x$ cm, of a plant and the number of leaves, $y$, on seven plants:$x$: $1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5$; $y$: $12, 15, 19, 22, 27, 29, 34$.(a) Write down the coordinates of the mean point $(\bar{x}, \bar{y})$.(b) Find the equation of the regression line of $y$ on $x$ in the form $y = ax + b$.(c) Interpret the value of $a$ in context.(d) Estimate the number of leaves on a plant of length $5$ cm and state one reason your estimate may be unreliable.(e) The scientist claims that a longer plant causes more leaves to grow. Comment on this claim.
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