Learn Extracted exam questions A-Level Mathematics 9709 Mathematics November 2025 Question Paper 62
9709 Mathematics November 2025 Question Paper 62
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The number, $X$, of used computers donated to a charity has a constant average rate of 2.4 computers per week.
State a necessary condition for $X$ to have a Poisson distribution.
Now assume that $X$ has a Poisson distribution. Calculate the probability that the number of computers donated during a 4-week period is more than 6 and less than 9.
Use a suitable approximating distribution to calculate the probability that more than 50 computers are donated during a 20-week period.
The random variable $X$ has a normal distribution with mean 10 and standard deviation 3. The independent random variable $Y$ has a Poisson distribution with mean 4.
Find the standard deviation of $X + Y$.
Find the standard deviation of $5X - Y$.
The times, in minutes, taken by students to complete a test have mean $\mu$ and standard deviation $\sigma$. The times taken by a random sample of 100 students are noted and are used to calculate a 95% confidence interval for $\mu$.
Given that the end points of the 95% confidence interval are 31.02 and 33.98, correct to 4 significant figures, calculate the value of $\sigma$.
The calculation of the confidence interval required the use of the Central Limit theorem. Explain why it is valid to use the Central Limit theorem in this case.
A researcher calculates a number, $r$, of 95% confidence intervals for $\mu$. Find the largest value of $r$ such that the probability that all $r$ confidence intervals contain the true value of $\mu$ is greater than 0.5.
An inspector believes that 18% of cups made at a certain factory contain flaws. The factory owner claims that the true percentage is less than 18%. The inspector examines a random sample of 40 cups and finds that 3 of them contain flaws.
Stating a necessary assumption, use a binomial distribution to test the factory owner's claim at the 5% significance level.
Explain why it would not be appropriate to use the Poisson approximation to the binomial distribution to carry out the test in part (a).
A random variable $X$ has probability density function given by $f(x) = k(2x^2 - x^3)$ for $0 \leqslant x \leqslant 2$, and $f(x) = 0$ otherwise.
Show that $k = \tfrac{3}{4}$.
The median of $X$ is denoted by $m$.
Write down the value of $P(X \leqslant m)$.
Hence find $P(E(X) \leqslant X \leqslant m)$.
The weekly profit, in dollars, made by a certain firm has a normal distribution. In the past, the weekly profit had the distribution $N(736, 26^2)$. Following a change in management, the mean weekly profit for 35 randomly chosen weeks is $725.
Stating a necessary assumption, test at the 2% significance level whether the mean weekly profit has decreased.
The mean weekly profit for another random sample of 35 weeks is found and a similar test is carried out at the 2% significance level. State the probability of a Type I error.
Given that the mean weekly profit is now in fact $718, find the probability of a Type II error.