Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 44
9231 Mathematics - Further November 2025 Question Paper 44
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Local residents are concerned about the speed of cars travelling through their village. They record the speed, in $\text{km h}^{-1}$, of a random sample of 13 cars travelling through their village. The recorded speeds are as follows: 40, 53, 59, 42, 43, 48, 62, 67, 46, 82, 66, 45, 70.
Construct a 90% confidence interval for the population mean speed of cars passing through the village.
State an assumption that is necessary for the confidence interval found in part (a) to be valid.
The continuous random variable $X$ has probability density function f given by $\mathrm{f}(x) = \tfrac49(x+1)$ for $0 \leqslant x \leqslant 1$, $(x-2)^2$ for $1 < x \leqslant 2$, and $0$ otherwise.
Find the cumulative distribution function of $X$.
Find the exact value of the median of $X$.
The random variable $Y$ is defined by $Y = \sqrt X$. Find the cumulative distribution function of $Y$.
Determine whether the median of $Y$ is greater than, or less than, the median of $X$.
Two different types of juice extractor, machine $X$ and machine $Y$, are being compared. The manufacturer claims that machine $Y$ extracts more juice per orange on average than machine $X$. A random sample of 20 oranges is selected, paired by similar size and mass and numbered 1 to 10; one orange from each pair is randomly allocated to machine $X$ and the other to machine $Y$. The amount of juice, in ml, extracted is recorded (pairs 1--10): machine $X$: 65, 73, 58, 61, 72, 79, 64, 65, 69, 71; machine $Y$: 68, 72, 64, 63, 75, 82, 63, 63, 72, 74.
Use a $t$-test to test the manufacturer's claim at the 1% significance level.
State an assumption required for the test in part (a) to be valid.
The manufacturer notices that the amount of juice extracted from the oranges in pair 4 were recorded incorrectly. In fact, the amount of juice extracted from machine $X$ was 60 ml and from machine $Y$ was 62 ml. Explain, with justification, whether the conclusion of the test in part (a) remains the same.
The random variable $X$ takes values 1 and 2 with probabilities $\tfrac25$ and $\tfrac35$ respectively.
Write down the probability generating function $\mathrm{G}_X(t)$ of $X$.
The random variable $Y$ is the sum of four independent observations of $X$. Find the probability generating function $\mathrm{G}_Y(t)$ of $Y$. Give your answer in the form $\mathrm{G}_Y(t) = at^m(b+ct)^n$, where $a$, $b$, $c$, $m$ and $n$ are constants to be determined.
Use $\mathrm{G}_Y(t)$ to find $\mathrm{P}(Y = 6)$.
Find $\mathrm{Var}(Y)$.
A driving instructor believes that the performance (pass or fail) of students when taking a driving test is associated with their age. The following table summarises the number of students who pass and who fail, and the ages in years of the students taking the test, over a period of three years. Pass: under 20 = 34, 20--40 = 41, over 40 = 6 (total 81). Fail: under 20 = 16, 20--40 = 39, over 40 = 9 (total 64). Column totals: 50, 80, 15 (grand total 145). Test, at the 10% significance level, whether performance is independent of age of student.
Students of the same age from two schools, school $A$ and school $B$, take a large number of quizzes throughout the year and are each awarded a mark out of 1000. The marks of 123 students in school $A$ and 147 students in school $B$ are ranked from lowest (rank 1) to highest (rank 270). The sum of the ranks of the students from school $A$ is 15355.
Carry out a Wilcoxon rank-sum test at the 5% significance level to investigate whether there is a difference in average marks between the students in school $A$ and school $B$.
State an assumption that is required for the Wilcoxon rank-sum test to be valid.