Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 42
9231 Mathematics - Further November 2025 Question Paper 42
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A large company claims that the median salary of its employees is $32500. The salaries ($) of 15 randomly selected employees are: 18750, 30500, 125000, 42500, 25000, 26000, 52500, 23000, 27500, 19500, 25500, 33000, 30000, 21500, 29000.
Explain why a Wilcoxon signed-rank test may not be appropriate to test the company's claim in this case.
Carry out a sign test at the 10% significance level to investigate the company's claim.
A factory produces packets of biscuits. The total mass of biscuits in a packet has a normal distribution with mean $\mu$. A random sample of 12 packets is taken and the mass of the contents of each packet, $x$ g, is recorded, giving $\sum x = 2390$ and $\sum x^2 = 476117$.
Find a 99% confidence interval for $\mu$.
A test of the null hypothesis $\mu = k$ is carried out on this sample using a 5% significance level. The test does not support the alternative hypothesis $\mu < k$. Find the greatest possible value of $k$.
A traffic expert claims that the number of breakdowns occurring each day on a busy section of motorway follows a Poisson distribution with mean 0.7. Over a 200-day period the observed frequencies for 0, 1, 2, 3, 4, $\geqslant 5$ breakdowns were 88, 73, 26, 7, 3, 3, with expected frequencies 99.317, $m$, 24.333, 5.678, 0.994, $n$.
Find the value of $m$ and the value of $n$, correct to 3 decimal places.
Carry out a goodness of fit test at the 5% significance level to investigate the expert's claim.
A continuous random variable $X$ has cumulative distribution function $F$ given by $F(x) = 0$ for $x < 1$, $F(x) = \tfrac15 x + a$ for $1 \leqslant x < 4$, $F(x) = \tfrac{1}{50}x^2 + b$ for $4 \leqslant x \leqslant 6$, and $F(x) = 1$ for $x > 6$, where $a$ and $b$ are constants.
Find the value of $a$ and the value of $b$.
Find the probability density function of $X$.
Given that $E(X) = \tfrac{529}{150}$, find $\mathrm{Var}(X)$.
Find the 10th and 90th percentiles of $X$.
An engineer is comparing the tensile strengths of steel rods made from two machines, $A$ and $B$, randomly selecting 8 rods from $A$ and 6 from $B$. Machine $A$: 402, 403, 415, 412, 409, 407, 406, 410. Machine $B$: 401, 398, 395, 397, 410, 405. Assuming the two distributions are normal with the same population variance, use a $t$-test at the 5% significance level to test whether there is any difference in the mean tensile strengths of steel rods from the two machines.
The discrete random variable $X$ has probability generating function $G_X(t) = \dfrac{t}{(3 - 2t)^2}$.
Find $E(X)$ and $\mathrm{Var}(X)$.
The discrete random variable $Y$ has probability generating function $G_Y(t) = \dfrac{t^2}{(3 - 2t)^2}$. The random variable $Z$ is the sum of $X$ and $Y$. Assuming $X$ and $Y$ are independent, find $P(Z > 4)$.