Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 41
9231 Mathematics - Further November 2025 Question Paper 41
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A group of 10 school children are asked to estimate the size of an angle $\theta\degree$ in a given acute angled triangle. These estimates, in degrees, are as follows: 84, 85, 77, 85, 84, 87, 86, 88, 83, 85.
Stating any assumptions you make, calculate a 95% confidence interval for $\theta$.
Give a reason why the assumptions made in part (a) may not be appropriate in this case.
The manager of a car park claims that the number of cars entering the car park follows a Poisson distribution with mean 2.8. The numbers of cars entering the car park are recorded on a working day during successive 5-minute periods. The table contains the observed frequencies, together with most of the expected frequencies and their contributions to the $\chi^2$-test statistic (with $p$, $q$ the missing expected frequency and contribution for 3 cars).
Find the value of $p$ and the value of $q$.
Carry out a goodness of fit test at the 5% significance level to investigate the manager's claim.
A random sample of 10 newborn baby boys is taken and their masses in kg are recorded. From this sample, the population standard deviation of all newborn baby boys is estimated as 0.6 kg. A random sample of 5 newborn baby girls is taken and their masses in kg are recorded as follows: 3.9, 3.1, 2.9, 3.1, 3.6. It is assumed that the masses of newborn baby boys and girls have the same population standard deviation, $\sigma$ kg. By pooling the two samples, calculate an estimate of $\sigma$.
A researcher believes that the median $m$ of a population has changed from its known previous value $m_0$. The researcher collects a random sample of size 28. She ranks the data and calculates a test statistic $T$ using the Wilcoxon signed-rank test. The conclusion of the test carried out at a 1% significance level is that there is not sufficient evidence to support her belief. Using a normal approximation, find the least possible value of $T$.
A continuous random variable $X$ has probability density function f given by $\mathrm{f}(x) = \tfrac{1}{16}\sqrt x$ for $0 \leqslant x < 4$, $\dfrac{1}{k\sqrt x}$ for $4 \leqslant x \leqslant 9$, and $0$ otherwise, where $k$ is a constant.
Show that $k = 3$.
Find the median value of $X$.
The random variable $Y$ is defined by $Y = \sqrt X$. Find the probability density function of $Y$.
Nine athletes in a club have a new coach. The coach adopts a new training programme which he believes will reduce the race times of these athletes. Each athlete completes a 1500 m time trial before and after completing the new training programme. Their times, in seconds (s), are recorded (athletes $A$–$I$: before 250, 251, 252, 267, 276, 291, 310, 320, 335; after 245, 251, 253, 261, 275, 293, 302, 313, 320).
Carry out a paired $t$-test at the 5% significance level to test the coach's belief.
Further research suggests that the effects of the training programme tend to reduce the times of the slower athletes by more than those of the faster athletes. Suggest a reason why the paired $t$-test used in part (a) may not have been an appropriate test in this case.
Suggest a suitable alternative test that could have been used instead of a paired $t$-test.
A discrete random variable $X$ takes values $r = 0, 1, 2$ with probabilities $\mathrm{P}(X=r)$ given by $\mathrm{P}(X=0) = a$, $\mathrm{P}(X=1) = 2a$, $\mathrm{P}(X=2) = b$.
Write down the probability generating function of $X$, and use it to find an expression for $\mathrm{E}(X)$ in terms of $a$ and $b$.
Show that $\mathrm{Var}(X) = 2b + 2(a+b)(1 - 2a - 2b)$.
The random variable $Y$ is defined by $Y = X_1 + X_2 + X_3 + \cdots + X_{10}$ where $X_1, X_2, X_3, \ldots, X_{10}$ are ten independent observations of $X$. Using the probability generating function of $Y$, and your answer to part (a), show that $\mathrm{E}(Y) = 10\mathrm{E}(X)$.
For the case $b = 0$, define fully the distribution of $Y$.