Learn Extracted exam questions A-Level Mathematics 9709 Mathematics June 2025 Question Paper 61
9709 Mathematics June 2025 Question Paper 61
Source PDF on the left, extracted YAML on the right. Compare numbering, marks, options and text.
It is known that 1% of houses in a certain area have a wind turbine. A random sample of 400 houses in this area is chosen for a survey on domestic heating. The number of houses in the sample that have a wind turbine is denoted by $X$. Use a suitable approximating distribution to find $\mathrm{P}(X \leqslant 3)$.
The random variable $X$ has the distribution $B\!\left(8, \tfrac34\right)$. A random sample of 100 values of $X$ is chosen, and the sample mean, $\overline{X}$, is found.
Find $\mathrm{P}(\overline{X} > 6.2)$. You are not expected to use a continuity correction.
State why the Central Limit Theorem was needed in the calculation in part (a).
The time, $T$ minutes, for a certain daily bus journey is normally distributed. The bus company claims that the mean of $T$ is 45. A passenger believes that the mean of $T$ is actually greater than 45. She notes the times taken for this journey on a random sample of 60 days. The results are summarised below: $n = 60$, $\sum t = 2750$, $\sum t^2 = 127000$.
Calculate unbiased estimates of the population mean and variance.
Test the passenger's belief at the 5% significance level.
At an entertainment centre, the cost for using a particular video game is $0.40 per minute. The number of minutes for which people use the video game has mean 15 and variance 9.
Find the mean and variance of the amount people pay for using the video game.
Each day, 35 people independently use the video game. Find the mean and variance of the total amount paid by 35 people.
The random variables $W$ and $X$ have the independent distributions $\mathrm{Po}(1.2)$ and $\mathrm{Po}(2.3)$ respectively.
Find $\mathrm{P}(3 \leqslant W + X \leqslant 5)$.
The random variable $S$ is the sum of 100 independent values of $W$ and 200 independent values of $X$. Use a suitable approximation to find $\mathrm{P}(S > 600)$.
The random variable $Y$ has the distribution $\mathrm{Po}(\lambda)$, where $\lambda > 0$. It is given that $\tfrac52\mathrm{P}(Y=3) + \mathrm{P}(Y=4) = \mathrm{P}(Y=5)$. Find the value of $\lambda$.
A manufacturer of cell phones claims that 25% of students own a Pumpkin phone. Jeyeraj thinks that the proportion of students at his large college who own a Pumpkin phone is less than 25%. He plans to test the manufacturer's claim. He chooses a random sample of 30 students at his college. If the number of students who own a Pumpkin phone is less than 5, Jeyeraj will reject the manufacturer's claim.
State suitable hypotheses for the test.
Given that the true proportion of students at the college who own a Pumpkin phone is 10%, use a binomial distribution to find the probability of a Type II error.
At Florence's college, in a random sample of 40 students, it was found that 5 own a Pumpkin phone. Calculate an approximate 95% confidence interval for the proportion of students at Florence's college who own a Pumpkin phone.
$X$ is a random variable with probability density function given by $f(x) = 1 + \cos\pi x$ for $0 \leqslant x \leqslant 1$, and $f(x) = 0$ otherwise.
Show that $\mathrm{P}\!\left(X < \tfrac12\right) = \tfrac12 + \tfrac1\pi$.
Show that $\mathrm{E}(X) = \tfrac12 - \tfrac{2}{\pi^2}$.