Questions - CAIE S2 - A-Level Maths

Find an approximate $96 \%$ confidence interval for $p$.
From three further experiments, Jack finds a 90\% confidence interval, a 95\% confidence interval and a 99\% confidence interval for $p$.
Find the probability that exactly two of these confidence intervals contain the true value of $p$.

CAIE S2 2021 November Q4

4 A certain kind of firework is supposed to last for 30 seconds, on average, after it is lit. An inspector suspects that the fireworks actually last a shorter time than this, on average. He takes a random sample of 100 fireworks of this kind. Each firework in the sample is lit and the time it lasts is noted.

Give a reason why it is necessary to take a sample rather than testing all the fireworks of this kind.
It is given that the population standard deviation of the times that fireworks of this kind last is 5 seconds.
The mean time lasted by the 100 fireworks in the sample is found to be 29 seconds. Test the inspector's suspicion at the $1 \%$ significance level.
State with a reason whether the Central Limit theorem was needed in the solution to part (b).

CAIE S2 2021 November Q5

5 In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.

Find the probability that there are fewer than 3 typing errors in 10 randomly chosen pages.
Use an approximating distribution to find the probability that there are more than 50 typing errors in 200 randomly chosen pages.
In the same document, formatting errors occur at random and at a constant mean rate of 0.3 per page.
Find the probability that the total number of typing and formatting errors in 20 randomly chosen pages is between 8 and 11 inclusive.

CAIE S2 2021 November Q6

6 A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is less than $\frac { 1 } { 10 }$. In order to test his suspicion he notes the number of times the digit 0 occurs in 30 digits produced by the machine. He carries out a test at the $10 \%$ significance level.

State suitable null and alternative hypotheses.
Find the rejection region for the test.
State the probability of a Type I error.
It is now given that the machine actually produces a 0 once in every 40 digits, on average.
Find the probability of a Type II error.
Explain the meaning of a Type II error in this context.

CAIE S2 2021 November Q7

The probability density function of the random variable $X$ is given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where $k$ is a constant.
1. Show that $k = \frac { 3 } { 16 }$.
2. Find $\mathrm { E } ( X )$.
The random variable $Y$ has the following properties.
- $Y$ takes values between 0 and 5 only.
- The probability density function of $Y$ is symmetrical.
Given that $\mathrm { P } ( Y < a ) = 0.2$, find $\mathrm { P } ( 2.5 < Y < 5 - a )$ illustrating your method with a sketch on the axes provided.
\includegraphics[max width=\textwidth, alt={}, center]{cea87af9-4b2a-4297-91e9-4eb5744b9e48-11_369_837_621_694}
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE S2 2022 November Q1

1 The heights, in metres, of a random sample of 10 mature trees of a certain variety are given below.
$\begin{array} { l l l l l l l l l l } 5.9 & 6.5 & 6.7 & 5.9 & 6.9 & 6.0 & 6.4 & 6.2 & 5.8 & 5.8 \end{array}$ Find unbiased estimates of the population mean and variance of the heights of all mature trees of this variety.

CAIE S2 2022 November Q2

2 A spinner has five sectors, each printed with a different colour. Susma and Sanjay both wish to test whether the spinner is biased so that it lands on red on fewer spins than it would if it were fair. Susma spins the spinner 40 times. She finds that it lands on red exactly 4 times.

Use a binomial distribution to carry out the test at the $5 \%$ significance level.
Sanjay also spins the spinner 40 times. He finds that it lands on red $r$ times.
Use a binomial distribution to find the largest value of $r$ that lies in the rejection region for the test at the 5\% significance level.

CAIE S2 2022 November Q3

3 Drops of water fall randomly from a leaking tap at a constant average rate of 5.2 per minute.

Find the probability that at least 3 drops fall during a randomly chosen 30 -second period.
Use a suitable approximating distribution to find the probability that at least 650 drops fall during a randomly chosen 2-hour period.

CAIE S2 2022 November Q4

4 Each month a company sells $X \mathrm {~kg}$ of brown sugar and $Y \mathrm {~kg}$ of white sugar, where $X$ and $Y$ have the independent distributions $\mathrm { N } \left( 2500,120 ^ { 2 } \right)$ and $\mathrm { N } \left( 3700,130 ^ { 2 } \right)$ respectively.

Find the mean and standard deviation of the total amount of sugar that the company sells in 3 randomly chosen months.
The company makes a profit of $
) 1.50\( per kilogram of brown sugar sold and makes a loss of $\$ 0.20$ per kilogram of white sugar sold.
Find the probability that, in a randomly chosen month, the total profit is less than \(
) 3000$.

CAIE S2 2022 November Q5

5 A builders' merchant sells stones of different sizes.

The masses of size $A$ stones have standard deviation 6 grams. The mean mass of a random sample of 200 size $A$ stones is 45 grams. Find a 95\% confidence interval for the population mean mass of size $A$ stones.
The masses of size $B$ stones have standard deviation 11 grams. Using a random sample of size 200, an $\alpha \%$ confidence interval for the population mean mass is found to have width 4 grams. Find $\alpha$.

CAIE S2 2022 November Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable $X$ that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line $x = 1$. Between $x = - 1$ and $x = 3$ the graph is a quadratic curve. The random variable $S$ is such that $\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )$ and $\operatorname { Var } ( S ) = \operatorname { Var } ( X )$.

On the grid below, sketch a quadratic graph for the probability density function of $S$.
\includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1191_1169_479} The random variable $T$ is such that $\mathrm { E } ( T ) = \mathrm { E } ( X )$ and $\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )$.
On the grid below, sketch a quadratic graph for the probability density function of $T$.
\includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
Given that $\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5$, show that $a ^ { 3 } - 12 a + 8 = 0$.
Hence verify that $0.69 < a < 0.70$.

CAIE S2 2022 November Q7

7 In the past Laxmi's time, in minutes, for her journey to college had mean 32.5 and standard deviation 3.1. After a change in her route, Laxmi wishes to test whether the mean time has decreased. She notes her journey times for a random sample of 50 journeys and she finds that the sample mean is 31.8 minutes. You should assume that the standard deviation is unchanged.

Carry out a hypothesis test, at the $8 \%$ significance level, of whether Laxmi's mean journey time has decreased.
Later Laxmi carries out a similar test with the same hypotheses, at the $8 \%$ significance level, using another random sample of size 50 .
Given that the population mean is now 31.5, find the probability of a Type II error.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE S2 2022 November Q1

1 Each of a random sample of 80 adults gave an estimate, $h$ metres, of the height of a particular building. The results were summarised as follows. $$n = 80 \quad \Sigma h = 2048 \quad \Sigma h ^ { 2 } = 52760$$

Calculate unbiased estimates of the population mean and variance.
Using this sample, the upper boundary of an $\alpha \%$ confidence interval for the population mean is 26.0 . Find the value of $\alpha$.

CAIE S2 2022 November Q2

2 In the past, the mean length of a particular variety of worm has been 10.3 cm , with standard deviation 2.6 cm . Following a change in the climate, it is thought that the mean length of this variety of worm has decreased. The lengths of a random sample of 100 worms of this variety are found and the mean of this sample is found to be 9.8 cm . Assuming that the standard deviation remains at 2.6 cm , carry out a test at the $2 \%$ significance level of whether the mean length has decreased.
$31.6 \%$ of adults in a certain town ride a bicycle. A random sample of 200 adults from this town is selected.

Use a suitable approximating distribution to find the probability that more than 3 of these adults ride a bicycle.
Justify your approximating distribution.

CAIE S2 2022 November Q4

4 The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per $10 \mathrm {~m} ^ { 2 }$. An adjustment is made to the machine. It is required to test at the $5 \%$ significance level whether the mean number of faults has decreased. A randomly selected $30 \mathrm {~m} ^ { 2 }$ of cloth is checked and the number of faults is found.

State suitable null and alternative hypotheses for the test.
Find the probability of a Type I error.
Exactly 3 faults are found in the randomly selected $30 \mathrm {~m} ^ { 2 }$ of cloth.
Carry out the test at the $5 \%$ significance level.
Later a similar test was carried out at the $5 \%$ significance level, using another randomly selected $30 \mathrm {~m} ^ { 2 }$ of cloth.
Given that the number of faults actually has a Poisson distribution with mean 0.5 per $10 \mathrm {~m} ^ { 2 }$, find the probability of a Type II error.
$5 X$ is a random variable with distribution $\mathrm { B } ( 10,0.2 )$. A random sample of 160 values of $X$ is taken.
Find the approximate distribution of the sample mean, including the values of the parameters.
Hence find the probability that the sample mean is less than 1.8 .

CAIE S2 2022 November Q6

6 The masses, in grams, of small and large bags of flour have the distributions $\mathrm { N } ( 510,100 )$ and $\mathrm { N } ( 1015,324 )$ respectively. André selects 4 small bags of flour and 2 large bags of flour at random.

Find the probability that the total mass of these 6 bags of flour is less than 4130 g .
Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags.

CAIE S2 2022 November Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{08f5a515-2c63-4955-b9da-7000631ff012-12_433_1006_255_568} The diagram shows the graph of the probability density function, f , of a random variable $X$ which takes values between - 3 and 2 only.

Given that the graph is symmetrical about the line $x = - 0.5$ and that $\mathrm { P } ( X < 0 ) = p$, find $\mathrm { P } ( - 1 < X < 0 )$ in terms of $p$.
It is now given that the probability density function shown in the diagram is given by $$f ( x ) = \begin{cases} a - b \left( x ^ { 2 } + x \right) & - 3 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where $a$ and $b$ are positive constants.
1. Show that $30 a - 55 b = 6$.
2. By substituting a suitable value of $x$ into $\mathrm { f } ( x )$, find another equation relating $a$ and $b$ and hence determine the values of $a$ and $b$.
  If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE S2 2022 November Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable $X$ that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line $x = 1$. Between $x = - 1$ and $x = 3$ the graph is a quadratic curve. The random variable $S$ is such that $\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )$ and $\operatorname { Var } ( S ) = \operatorname { Var } ( X )$.

On the grid below, sketch a quadratic graph for the probability density function of $S$.
\includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1191_1169_479} The random variable $T$ is such that $\mathrm { E } ( T ) = \mathrm { E } ( X )$ and $\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )$.
On the grid below, sketch a quadratic graph for the probability density function of $T$.
\includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
Given that $\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5$, show that $a ^ { 3 } - 12 a + 8 = 0$.
Hence verify that $0.69 < a < 0.70$.

CAIE S2 2023 November Q1

1 A random variable $X$ has the distribution $\mathrm { N } ( 410,400 )$.
Find the probability that the mean of a random sample of 36 values of $X$ is less than 405 .

CAIE S2 2023 November Q2

2 In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an $\alpha \%$ confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487 , correct to 3 significant figures. Find the value of $\alpha$ correct to the nearest integer.

CAIE S2 2023 November Q3

3 A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.

Assume that the owner is correct.
1. Find the probability that there will be at least 4 hits during a 10-minute period.
2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period.
  A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time ( 9.00 am to 9.00 pm ) is usually about twice the number of hits during the night-time ( 9.00 pm to 9.00 am ).
1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution.
2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time.

CAIE S2 2023 November Q4

4 The masses, in kilograms, of chemicals $A$ and $B$ produced per day by a factory are modelled by the independent random variables $X$ and $Y$ respectively, where $X \sim \mathrm {~N} ( 10.3,5.76 )$ and $Y \sim \mathrm {~N} ( 11.4,9.61 )$. The income generated by the chemicals is $
) 2.50\( per kilogram for $A$ and $\$ 3.25$ per kilogram for $B$.

Find the mean and variance of the daily income generated by chemical $A$.
\includegraphics[max width=\textwidth, alt={}, center]{d42b3c4d-c426-4231-a35a-cac80dbdf82c-06_56_1566_495_333}
Find the probability that, on a randomly chosen day, the income generated by chemical $A$ is greater than the income generated by chemical $B$.

CAIE S2 2023 November Q5

5 In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution $\operatorname { Po } ( 0.31 )$. Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5 , carry out the test at the $2.5 \%$ significance level.

CAIE S2 2023 November Q6

6 A continuous random variable $X$ takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, $a$ and $b$, of $X$ are such that $\mathrm { P } ( a < X < b ) = p$ and $\mathrm { P } ( b < X < 3 ) = \frac { 13 } { 10 } p$, where $p$ is a positive constant.

Show that $p \leqslant \frac { 5 } { 23 }$.
Find $\mathrm { P } ( b < X < 6 - a )$ in terms of $p$.
It is now given that the probability density function of $X$ is f , where $$f ( x ) = \begin{cases} \frac { 1 } { 36 } \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$
Given that $b = 2$ and $p = \frac { 5 } { 27 }$, find the value of $a$.

CAIE S2 2023 November Q7

7 A biologist wishes to test whether the mean concentration $\mu$, in suitable units, of a certain pollutant in a river is below the permitted level of 0.5 . She measures the concentration, $x$, of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. $$n = 50 \quad \Sigma x = 23.0 \quad \Sigma x ^ { 2 } = 13.02$$

Carry out a test at the $5 \%$ significance level of the null hypothesis $\mu = 0.5$ against the alternative hypothesis $\mu < 0.5$.
Later, a similar test is carried out at the $5 \%$ significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
Given that, in fact, the value of $\mu$ is 0.4 , find the probability of a Type II error.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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