Every question, cut by section

Which of these best describes what a mathematical model is, in the sense spec item 1.1 means it?

X ~ N(170, 625). Show that P(X > 196) = 0.15, correct to 2 significant figures. (3 marks)

Which response below is certain to score all 3 marks?

A question needs the z-value satisfying $P(Z > z) = 0.05$. Which value should be written down?

$X \sim N(30, 4^2)$. A student wants $P(X < 34)$. After standardising to $z = 1$ and reading $\Phi(1) = 0.8413$ from the table, what should they do next?

A discrete random variable $X$ has $E(X) = 12$ and $\text{Var}(X) = 5$. Given that $Y = 3X - 4$, find $E(Y)$ and $\text{Var}(Y)$.

A discrete random variable $X$ has $E(X) = 5$ and $\text{Var}(X) = 3$. The random variable $Y$ is defined by $Y = 2X + 1$. What is $\text{Var}(Y)$?

A discrete random variable $X$ has $p(1)=0.15$, $p(2)=0.35$, $p(3)=0.25$, $p(4)=0.25$. Find $F(3)$.

A discrete random variable $X$ can take the values $0, 1, 2$ and $3$ only. A student writes $p(1) = 0.2$, $p(2) = 0.4$, $p(3) = 0.3$ and stops, treating this as the complete probability function. What does checking $\sum p(x) = 1$ reveal?

For a different dataset, $n = 6$, $\sum x = 54$, $\sum y = 96$, $\sum x^2 = 522$, $\sum xy = 918$. What is the gradient $b$ of the regression line of $y$ on $x$?

A scatter diagram shows a clear U-shaped relationship between $x$ and $y$: as $x$ increases, $y$ first decreases, then increases. The PMCC for this data works out as $r = 0.04$. Which is the most defensible conclusion?

A calculation for $r$ involves dividing by a square root — for example, $90 \div \sqrt{15000}$ — and the final answer needs to be given to 3 significant figures. What's the safest amount of precision to use for the square root itself, before dividing?

A bag contains 5 counters: 2 red and 3 green. Two counters are drawn at random, WITH replacement. What is $P(\text{both red})$?

A Venn diagram for events A and B is drawn inside a rectangle representing the sample space. $P(A \text{ only}) = 0.3$, $P(B \text{ only}) = 0.25$ and $P(A \cap B) = 0.15$. What value belongs in the region of the rectangle outside both circles?

In a class, $P(\text{studies Chemistry}) = 0.5$, $P(\text{studies Biology}) = 0.6$, and $P(\text{studies both}) = 0.2$. Find $P(\text{Biology} \mid \text{NOT Chemistry})$.

Events $A$ and $B$ satisfy $P(A) = 0.3$, $P(B) = 0.4$, and $P(A \cap B) = 0$. What can you correctly conclude?

Let $N$ = 'a customer buys a newspaper' and $M$ = 'a customer buys a magazine'. Which expression correctly represents 'the probability a customer buys a magazine, GIVEN that they have already bought a newspaper'?

A data set has $Q_1 = 10$, $Q_3 = 25$. Using $1.5 \times IQR$, the upper fence is 47.5. The highest value in the data that is NOT an outlier is 40; one value, 55, is an outlier. Where does the upper whisker end, and where is 55 shown?

A box plot's upper fence, from the rule given in the question, works out at 50. The data's true maximum is 61, which the rule identifies as an outlier; the highest value that is NOT an outlier is 46. Where should the upper whisker actually end?

A grouped frequency table has $n = 39$. One student finds $Q_1$'s position using $n/4 = 9.75$; another uses $(n+1)/4 = 10$. Both interpolate correctly from their own position within whichever class it falls into. What's true of their two final answers?

For a frequency table with midpoints $x$, frequencies $f$, and $n = \Sigma f$, which expression correctly defines the standard deviation?

Group A (Oak saplings, $n = 11$), sorted: 12, 15, 19, 21, 23, 26, 28, 29, 34, 37, 39.

What is $Q_1$ for Group A?

In a back-to-back stem-and-leaf diagram, which leaf represents the SMALLEST value on the LEFT-hand side of a row?

In the real WST01 question this lesson's worked chain is built from (Jan 2025, Q1), a second die — blue, with faces numbered 1, 3, 5, 7 — is also assumed fair, and its score is modelled by $B$. Which of these correctly describes $B$'s distribution, and why?

A real WST01 question gives $S_{xy} = 12105.12$ and $S_{xx} = 16769.78$, and asks you to show that the regression line is $g = -42.3 + 0.722d$ (3 s.f.) — the actual Jun 2024 Q4(c).

What is the safest working practice for how many decimal places to carry the gradient $b = S_{xy}/S_{xx}$ to, before rounding down to the printed line?

A 'show that' question asks a student to show that $P(X > 24) = 0.15$ for some $X \sim N(\mu, \sigma^2)$.

Which response is certain to earn full marks?

$X \sim N(40, 6^2)$. Find $P(X < 46)$.

Which of the following would you need to recall from memory in a WST01 exam, rather than being able to look up in the Mathematical Formulae and Statistical Tables booklet?

A discrete random variable $X$ has $E(X) = 6$ and $E(X^2) = 40$. A student writes: "$\text{Var}(X) = E(X^2) = 40$." What is wrong with this, and what is the correct value of $\text{Var}(X)$?

Two fair four-sided spinners are used independently: spinner $P$ shows 1, 2, 3, 4 and spinner $Q$ shows 2, 4, 6, 8, each value equally likely (both are discrete uniform distributions). What is $P(P=2 \text{ and } Q=2)$?

A discrete random variable $X$ takes the values 1, 2, 3, 4 with $p(1)=0.1$, $p(2)=0.2$, $p(3)=0.3$, $p(4)=0.3$. A student checks that every one of these four numbers lies between 0 and 1, and concludes this is a valid probability function. Is the student's check sufficient?

A student correctly finds $S_{xx} = 25$ and $S_{xy} = 40$ for a dataset, and correctly finds $\bar x = 4$, $\bar y = 11$. Which of these is an acceptable way to state the final regression line?

Using $\hat y = 10.1 + 4.25x$, someone predicts the enquiries when $5000 is spent on advertising ($x = 50$), even though the data used to fit the line only ever recorded spending between $200 and $1000. What's the soundest judgement of that prediction?

A question asks: 'Find r, giving your answer to 3 significant figures,' and your working reaches the step $90 \div \sqrt{15000}$. What's the safest way to evaluate this?

A large, unwieldy dataset is coded (e.g. subtracting a fixed value and dividing by another) purely to make the arithmetic manageable before finding its PMCC. Which best describes how that coding affects the actual value of $r$?

A Venn diagram for a class of 60 students and events $A$ (studies French) and $B$ (studies Spanish) shows 18 students in $A$ only, 22 in $B$ only, and 12 in both. How many students are outside both circles, and how should that be shown on the diagram?

On a Venn diagram for events A and B, $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$. Are A and B independent?

Events $C$ and $D$ satisfy $P(C) = 0.6$, $P(D) = 0.5$, and $P(C \cap D) = 0.3$. Are $C$ and $D$ independent, mutually exclusive, both, or neither?

Three fair coins are tossed. By listing the sample space systematically, find $P(\text{exactly two heads})$.

In a college group, $P(\text{studies French}) = 0.55$, $P(\text{studies Spanish}) = 0.4$, $P(\text{studies both}) = 0.2$. Find $P(\text{studies French or Spanish, or both})$.

A box plot has $Q_1 = 15$, median $= 19$, $Q_3 = 21$, with whiskers reaching 5 (minimum) and 24 (maximum, not an outlier). What does this suggest about the skew of the distribution?

A data set contains one very large outlier. Why might the median be quoted as "a typical value" rather than the mean?

A frequency table has values 10, 20, 30, 40 with frequencies 3, 5, 7, 5 ($\Sigma f = 20$). What is the mean?

A data set $X$ is coded as $Y = 0.5(X - 100)$. Given $\text{Var}(Y) = 9$, what is $\text{Var}(X)$?

Class P: width 10, frequency density 2. Class Q: width 3, frequency density 5.

Which class contains MORE data values?

A histogram bar for the class $4 \leq x < 8$ (width 4) has height (frequency density) 5. What is the frequency for this class?

A question states "model the journey times as Normally distributed," then later shows a box plot of the same data with strong positive skew. What does this situation best illustrate?

Having correctly calculated the outlier boundaries as 8 and 52, a question asks you to "show that there are 3 outliers" in a dataset. What must your answer additionally include to be safe?

$M \sim N(25, 4^2)$. Given that $M > 20$, find $P(M > 28 \mid M > 20)$.

$M \sim N(\mu, \sigma^2)$. A question gives you that $M > 40$, and asks for $P(M > 55 \mid M > 40)$. What kind of calculation is this?

A company pays each employee a monthly bonus, in dollars, $B = 50N - 20$, where $N$ is the random number of new clients that employee brings in that month. Records show $E(N) = 4$ and $\text{Var}(N) = 1.5$. Find $E(B)$ and $\text{Var}(B)$.

A discrete random variable $X$ has $E(X) = 3$ and $\text{Var}(X) = 2$. The random variable $W$ is defined by $W = 5 - 2X$ (so $a = -2$, $b = 5$). What is $\text{Var}(W)$?

A discrete random variable $U$ is Discrete Uniform on $\{2,4,6\}$. Find $\text{Var}(U)$.

A discrete random variable $W$ is Discrete Uniform on the values $\{5, 10, 15, 20\}$ — these four values are equally likely. What is $p(10)$?

A regression line is fitted to data recorded for $x$ between 10 and 50. Which of these predictions is NOT extrapolation?

The least squares regression line for this data turns out to be $\hat y = 10.1 + 4.25x$, where $x$ is spend in hundreds of dollars. What does the number 4.25 actually measure?

A dataset is coded using $u = \dfrac{x-50}{5}$ and $v = \dfrac{y-20}{2}$ to simplify the arithmetic. The PMCC of the coded data, $r_{uv}$, is found to be $0.64$. What is $r$ for the original, uncoded $x$ and $y$ data?

You already know how to find $S_{xx}$ and $S_{xy}$ for a regression line. To find $r$, the product moment correlation coefficient, you need one more summary quantity. Which one, and why does the gradient $b$ never need it?

On a Venn diagram, $P(C) = 0.6$, $P(D) = 0.5$ and $P(C \cap D) = 0.2$. What is $P(C \mid D)$?

A box contains 5 discs: 3 black and 2 white. One disc is removed at random and not replaced. Given that the disc removed was black, what is the probability that a second disc, taken from what remains, is also black?

Two fair six-sided dice are rolled. $A$ = 'the first die shows a 6'. $B$ = 'the two dice sum to 7'. Are $A$ and $B$ independent?

A card is drawn from a standard 52-card deck (4 Kings, 4 Queens, 44 others). Given that the card is a King or a Queen, find the probability it is a King.