Learn Extracted exam questions A-Level Mathematics 9709 Mathematics November 2025 Question Paper 61
9709 Mathematics November 2025 Question Paper 61
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The random variables $X$ and $Y$ have independent distributions $X \sim \mathrm{Po}(3)$ and $Y \sim \mathrm{Po}(2)$ respectively.
Find $P(2 < X < 5)$.
Find $P(X + Y > 2)$.
The total of 100 random values of $X$ and 150 random values of $Y$ is denoted by $T$. Use a suitable approximating distribution to find $P(T < 560)$.
The mean mass of packets of Trueleaf tea is supposed to be 500 grams. An inspector weighs 60 randomly chosen packets, giving $n = 60$, $\sum x = 29970$, $\sum x^2 = 14970300$. Test, at the 5% significance level, whether the population mean mass is 500 grams.
The data produced by a certain data entry firm include a small number of incorrect characters that occur at random, with proportion $p$; experience shows $p = 0.0001$. A particular data set contains 14500 characters, of which $X$ are incorrect.
Use a suitable approximating distribution to find $P(X < 4)$.
The management wishes to decrease $p$ by training so that, for a data set of 14500 characters, $P(X = 0)$ for the new $p$ is double the value of $P(X = 0)$ when $p = 0.0001$. Use a suitable approximating distribution to find the new value of $p$.
The masses of a certain species of animal are normally distributed with standard deviation $\sigma$ kg. A researcher uses a random sample of $n$ animals to find two confidence intervals ($\alpha\%$ and 90%) for the population mean. The width of the $\alpha\%$ confidence interval is $1.414\times$ the width of the 90% confidence interval.
Find $\alpha$.
Find the probability that the 90% confidence interval contains the population mean given that the $\alpha\%$ confidence interval contains the population mean.
It is known that 20% of households in a certain country contain more than 4 people. Laxmi believes that in her town the percentage is lower than 20%. She takes a random sample of 40 households and notes the number containing more than 4 people, then carries out a test at the 2.5% significance level using a binomial distribution.
Find the probability of a Type I error.
State the rejection region for the test.
Laxmi finds that exactly 2 households in her sample contain more than 4 people. Explain why it is impossible for Laxmi to make a Type II error.
The masses, in kilograms, of large and small bags of potatoes have the independent distributions $N(2.5, 0.05)$ and $N(0.8, 0.02)$ respectively.
Find the probability that the total mass of a randomly chosen large bag and a randomly chosen small bag is more than 3.55 kg.
Find the probability that the mass of a randomly chosen large bag is less than 3 times the mass of a randomly chosen small bag.
The time, in minutes, taken by students to complete a test is modelled by the random variable $X$ with probability density function $f(x) = -\tfrac34(x-3)(x-5)$ for $3 \leqslant x \leqslant 5$, and $f(x) = 0$ otherwise.
Find the probability that a randomly chosen student takes longer than 4.5 minutes to complete the test.
Write down the median of $X$.
Without performing an integration, use your answer to part (a) to find $P(3.5 < X < 4.5)$.