Questions - Edexcel - A-Level Maths

Find the probability that the area of the circle is greater than $10 \mathrm {~cm} ^ { 2 }$
State, giving a reason, whether the median area of the circle is greater or less than $10 \mathrm {~cm} ^ { 2 }$ The triangle $O A B$ is drawn inside the circle with $O A$ and $O B$ as radii of length $x \mathrm {~cm}$ and angle $A O B x$ radians.
Use algebraic integration to find the expected value of the area of triangle $O A B$. Give your answer as an exact value.

Edexcel FS2 2021 June Q1

7 marks Standard +0.3

Anisa is investigating the relationship between marks on a History test and marks on a Geography test. She collects information from 7 students. She wants to calculate the Spearman's rank correlation coefficient for the 7 students so she ranks their performance on each test.

Student	History mark	Geography mark	History rank	Geography rank
A	76	58	1	3
B	70	60	2	2
C	64	57	$s$	$t$
D	64	63	$s$	1
E	64	57	$s$	$t$
F	59	50	6	7
G	55	52	7	6

Write down the value of $s$ and the value of $t$ The full product moment correlation coefficient (pmcc) formula is used with the ranks to calculate the Spearman's rank correlation coefficient instead of $r _ { s } = 1 - \frac { 6 \Sigma d ^ { 2 } } { n \left( n ^ { 2 } - 1 \right) }$ and the value obtained is 0.7106 to 4 significant figures.
Explain why the full pmcc formula is used to carry out the calculation.
Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher a student ranks in the History test, the higher the student ranks in the Geography test. Use a $5 \%$ level of significance.

Edexcel FS2 2021 June Q2

9 marks Standard +0.3

A company produces two colours of candles, blue and white. The standard deviation of the burning times of the blue candles is 2.6 minutes and the standard deviation of the burning times of the white candles is 2.4 minutes.

Nissim claims that the mean burning time of blue candles is more than 5 minutes greater than the mean burning time of white candles. A random sample of 90 blue candles is found to have a mean burning time of 39.5 minutes. A random sample of 80 white candles is found to have a mean burning time of 33.7 minutes.

Stating your hypotheses clearly, use a suitable test to assess Nissim's belief. Use a $1 \%$ level of significance.
Explain how the hypothesis test in part (a) would be carried out differently if the variances of the burning times of candles were unknown. The burning times for the candles may not follow a normal distribution.
Describe the effect this would have on the calculations in the hypothesis test in part (a). Give a reason for your answer.

Edexcel FS2 2021 June Q3

10 marks Standard +0.3

The continuous random variable $X$ has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 2 \\ 1.25 - \frac { 2.5 } { x } & 2 \leqslant x \leqslant 10 \\ 1 & x > 10 \end{array} \right.$$

Find $\mathrm { P } ( \{ X < 5 \} \cup \{ X > 8 \} )$
Find the median of $X$.
Find $\mathrm { E } \left( X ^ { 2 } \right)$
1. Sketch the probability density function of $X$.
2. Describe the skewness of the distribution of $X$.

Edexcel FS2 2021 June Q4

10 marks Standard +0.3

A researcher is investigating the relationship between elevation, $x$ metres, and annual mean temperature, $t ^ { \circ } \mathrm { C }$.

From a random sample of 20 weather stations in Switzerland, the following results were obtained $$\mathrm { S } _ { x x } = 8820655 \quad \mathrm {~S} _ { t t } = 444.7 \quad \sum x = 28130 \quad \sum t = 94.62$$ The product moment correlation coefficient for these data is found to be - 0.959

Interpret the value of this correlation coefficient.
Show that the equation of the regression line of $t$ on $x$ can be written as $$t = 14.3 - 0.00681 x$$ The random variable $W$ represents the elevations of the weather stations in kilometres.
Write down the equation of the regression line of $t$ on $w$ for these 20 weather stations in the form $t = a + b w$
Show that the residual sum of squares (RSS) for the model for $t$ and $x$ is 35.7 correct to one decimal place. One of the weather stations in the sample had a recorded elevation of 1100 metres and an annual mean temperature of $1.4 ^ { \circ } \mathrm { C }$
1. Calculate this weather station's contribution to the residual sum of squares. Give your answer as a percentage
2. Comment on the data for this weather station in light of your answer to part (e)(i).

Edexcel FS2 2021 June Q5

10 marks Challenging +1.2

The continuous random variable $X$ is uniformly distributed over the interval $[ 0,4 \beta ]$, where $\beta$ is an unknown constant.

Three independent observations, $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$, are taken of $X$ and the following estimators for $\beta$ are proposed $$\begin{aligned} & A = \frac { X _ { 1 } + X _ { 2 } } { 2 } \\ & B = \frac { X _ { 1 } + 2 X _ { 2 } + 3 X _ { 3 } } { 8 } \\ & C = \frac { X _ { 1 } + 2 X _ { 2 } - X _ { 3 } } { 8 } \end{aligned}$$

Calculate the bias of $A$, the bias of $B$ and the bias of $C$
By calculating the variances, explain which of $B$ or $C$ is the better estimator for $\beta$
Find an unbiased estimator for $\beta$

Edexcel FS2 2021 June Q6

15 marks Challenging +1.2

Elsa is collecting information on the wingspan of two different species of butterfly, Ringlet and Meadow Brown. She takes a random sample of each type of butterfly. The wingspans, $w \mathrm {~cm}$, are summarised in the table below. The wingspans of Ringlet and Meadow Brown butterflies each follow normal distributions.

Test, at the $2 \%$ level of significance, whether or not there is evidence that the variance of the wingspans of Ringlet butterflies is different from the variance of the wingspans of Meadow Brown butterflies. You should state your hypotheses clearly. The $k \%$ confidence interval for the variance of the wingspans of Meadow Brown butterflies is (1.194, 48.54)
Find the value of $k$
Calculate a $95 \%$ confidence interval for the difference between the mean wingspan of the Ringlet butterfly and the mean wingspan of the Meadow Brown butterfly.

Edexcel FS2 2021 June Q7

14 marks Standard +0.8

The weights of a particular type of apple, $A$ grams, and a particular type of orange, $R$ grams, each follow independent normal distributions.

$$A \sim \mathrm {~N} \left( 160,12 ^ { 2 } \right) \quad R \sim \mathrm {~N} \left( 140,10 ^ { 2 } \right)$$

Find the distribution of
1. $A + R$
2. the total weight of 2 randomly selected apples. A box contains 4 apples and 1 orange only. Jesse selects 2 pieces of fruit at random from the box.
Find the probability that the total weight of the 2 pieces of fruit exceeds 310 grams. From a large number of apples and oranges, Celeste selects $m$ apples and 1 orange at random. The random variable $W$ is given by $$W = \left( \sum _ { i = 1 } ^ { m } A _ { i } \right) - n \times R$$ where $n$ is a positive integer.
Given that the middle $95 \%$ of the distribution of $W$ lies between 1100.08 and 1499.92 grams, (c) find the value of $m$ and the value of $n$

Edexcel FS2 2022 June Q1

7 marks Standard +0.3

Kwame is investigating a possible relationship between average March temperature, $t ^ { \circ } \mathrm { C }$, and tea yield, $y \mathrm {~kg} /$ hectare, for tea grown in a particular location. He uses 30 years of past data to produce the following summary statistics for a linear regression model, with tea yield as the dependent variable.

$$\begin{aligned} & \text { Residual Sum of Squares } ( \mathrm { RSS } ) = 1666567 \quad \mathrm {~S} _ { t t } = 52.0 \quad \mathrm {~S} _ { y y } = 1774155 \\ & \text { least squares regression line: } \quad \text { gradient } = 45.5 \quad y \text {-intercept } = 2080 \end{aligned}$$

Use the regression model to predict the tea yield for an average March temperature of $20 ^ { \circ } \mathrm { C }$ He also produces the following residual plot for the data.
\includegraphics[max width=\textwidth, alt={}, center]{d139840b-16ec-42ce-8501-f79c263c8017-02_663_880_868_589}
Explain what you understand by the term residual.
Calculate the product moment correlation coefficient between $t$ and $y$
Explain why the linear model may not be a good fit for the data
1. with reference to your answer to part (c)
2. with reference to the residual plot. \section*{Question 1 continues on page 4} Kwame also collects data on total March rainfall, $w \mathrm {~mm}$, for each of these 30 years. For a linear regression model of $w$ on $t$ the following summary statistic is found. $$\text { Residual Sum of Squares (RSS) = } 86754$$ Kwame concludes that since this model has a smaller RSS, there must be a stronger linear relationship between $w$ and $t$ than between $y$ and $t$ (where RSS $= 1666567$ )
State, giving a reason, whether or not you agree with the reasoning that led to Kwame's conclusion.

Edexcel FS2 2022 June Q2

12 marks Standard +0.3

A factory produces yellow tennis balls and white tennis balls. Independent samples, one of yellow tennis balls and one of white tennis balls, are taken. The table shows information about the weights of the yellow tennis balls, $Y$ grams, and the weights of the white tennis balls, $W$ grams.

	Sample size	Mean weight of random sample (grams)	Known population standard deviation of weights (grams)
Yellow tennis balls	120	57.2	1.2
White tennis balls	140	56.9	0.9

Find a 95\% confidence interval for the mean weight of yellow tennis balls. Jamie claims that the mean weight of the population of yellow tennis balls is greater than the mean weight of the population of white tennis balls. A test of Jamie's claim is carried out.
1. Specify the approximate distribution of $\bar { Y } - \bar { W }$ under the null hypothesis of the test.
2. Explain the relevance of the large sample sizes to your answer to part (i).
Complete the hypothesis test using a $5 \%$ level of significance. You should state your hypotheses and the value of your test statistic clearly.

Edexcel FS2 2022 June Q3

6 marks Standard +0.8

The random variable $X \sim \mathrm {~N} \left( 5,0.4 ^ { 2 } \right)$ and the random variable $Y \sim \mathrm {~N} \left( 8,0.1 ^ { 2 } \right)$
$X$ and $Y$ are independent random variables.
A random sample of $a$ independent observations is taken from the distribution of $X$ and one observation is taken from the distribution of $Y$

The random variable $W = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { a } + b Y$ and has the distribution $\mathrm { N } \left( 169,2 ^ { 2 } \right)$
Find the value of $a$ and the value of $b$

Edexcel FS2 2022 June Q4

8 marks Standard +0.8

A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.

Patient	$A$	$B$	C	$D$	$E$	$F$	$G$
Resting heart rate before	65	68	77	79	80	88	92
Resting heart rate after	63	65	73	76	80	84	80

Using a $5 \%$ level of significance, carry out an appropriate test of the doctor's belief. You should state your hypotheses, test statistic and critical value.
State the assumption made about the resting heart rates that was required to carry out the test.

Edexcel FS2 2022 June Q5

8 marks Standard +0.8

The concentration of an air pollutant is measured in micrograms $/ \mathrm { m } ^ { 3 }$

Samples of air were taken at two different sites and the concentration of this particular air pollutant was recorded. For Site $A$ the summary statistics are shown below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	number of samples	$S _ { A } ^ { 2 }$
Site $A$	13	6.39

For Site $B$ there were 9 samples of air taken.
A test of the hypothesis $\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }$ against the hypothesis $\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }$ is carried out using a $2 \%$ level of significance.

State a necessary assumption required to carry out the test. Given that the assumption in part (a) holds,
find the set of values of $s _ { B } ^ { 2 }$ that would lead to the null hypothesis being rejected,
find a 99\% confidence interval for the variance of the concentration of the air pollutant at Site A.

Edexcel FS2 2022 June Q6

15 marks Challenging +1.8

Korhan and Louise challenge each other to find an estimator for the mean, $\mu$, of the continuous random variable $X$ which has variance $\sigma ^ { 2 }$
$X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }$ are $n$ independent observations taken from $X$
Korhan's estimator is given by

$$K = \frac { 2 } { n ( n + 1 ) } \sum _ { r = 1 } ^ { n } r X _ { r }$$ Louise's estimator is given by $$L = \frac { X _ { 1 } + X _ { 2 } } { 3 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 3 ( n - 2 ) }$$

Show that $K$ and $L$ are both unbiased estimators of $\mu$
1. Find $\operatorname { Var } ( K )$
2. Find $\operatorname { Var } ( L )$ The winner of the challenge is the person who finds the better estimator.
Determine the winner of the challenge for large values of $n$. Give reasons for your answer.

Edexcel FS2 2022 June Q7

7 marks Challenging +1.2

A rectangle is to have an area of $40 \mathrm {~cm} ^ { 2 }$

The length of the rectangle, $L \mathrm {~cm}$, follows a continuous uniform distribution over the interval [4, 10] Find the expected value of the perimeter of the rectangle.
Use algebraic integration, rather than your calculator, to evaluate any definite integrals.

Edexcel FS2 2022 June Q8

12 marks Standard +0.3

The continuous random variable $X$ has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 1 \\ 1.5 x - 0.25 x ^ { 2 } - 1.25 & 1 \leqslant x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$

Find the exact value of the median of $X$
Find $\mathrm { P } ( X < 1.6 \mid X > 1.2 )$ The random variable $Y = \frac { 1 } { X }$
Specify fully the cumulative distribution function of $Y$
Hence or otherwise find the mode of $Y$

Edexcel FS2 2023 June Q1

7 marks Easy -1.2

Baako is investigating the times taken by children to run a 100 m race, $x$ seconds, and a 500 m race, $y$ seconds. For a sample of 20 children, Baako obtains the time taken by each child to run each race.

Here are Baako's summary statistics. $$\begin{gathered} \mathrm { S } _ { x x } = 314.55 \quad \mathrm {~S} _ { y y } = 9026 \quad \mathrm {~S} _ { x y } = 1610 \\ \bar { x } = 19.65 \quad \bar { y } = 108 \end{gathered}$$

Calculate the product moment correlation coefficient between the times taken to run the 100 m race and the times taken to run the 500 m race.
Show that the equation of the regression line of $y$ on $x$ can be written as $$y = 5.12 x + 7.42$$ where the gradient and $y$ intercept are given to 3 significant figures. The child who completed the 100 m race in 20 seconds took 104 seconds to complete the 500 m race.
Find the residual for this child. The table below shows the signs of the residuals for the 20 children in order of finishing time for the 100 m race.
Sign of residual + + + + - - + - - - - - - - - + + + + +
Explain what the signs of the residuals show about the model's predictions of the 500 m race times for the children who are fastest and slowest over the 100 m race.

Edexcel FS2 2023 June Q2

12 marks Standard +0.3

Camilo grows two types of apple, green apples and red apples.

The standard deviation of the weights of green apples is known to be 3.5 grams.
A random sample of 80 green apples has a mean weight of 128 grams.

Find a 98\% confidence interval for the mean weight of the population of green apples. Show your working clearly and give the confidence interval limits to 2 decimal places. Camilo believes that the mean weight of the population of green apples is more than 10 grams greater than the mean weight of the population of red apples. A random sample of $n$ red apples has a mean weight of 117 grams.
The standard deviation of the weights of the red apples is known to be 4 grams.
A test of Camilo's belief is carried out at the 5\% level of significance.
State the null and alternative hypotheses for this test.
Find the smallest value of $n$ for which the null hypothesis will be rejected.
Explain the relevance of the Central Limit Theorem in parts (a) and (c).
Given that $n = 85$, state the conclusion of the hypothesis test.

Edexcel FS2 2023 June Q3

8 marks Challenging +1.2

Two machines, $A$ and $B$, are used to fill bottles of water. The amount of water dispensed by each machine is normally distributed.

Samples are taken from each machine and the amount of water, $x \mathrm { ml }$, dispensed in each bottle is recorded. The table shows the summary statistics for Machine $A$.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Sample size	$\sum x$	$\sum x ^ { 2 }$
Machine $A$	9	2268	571700

Find a 95\% confidence interval for the variance of the amount of water dispensed in each bottle by Machine $A$. For Machine $B$, a random sample of 11 bottles is taken. The sample variance of the amount of water dispensed in bottles is $12.7 \mathrm { ml } ^ { 2 }$
Test, at the $10 \%$ level of significance, whether there is evidence that the variances of the amounts of water dispensed in bottles by the two machines are different. You should state the hypotheses and the critical value used.

Edexcel FS2 2023 June Q4

8 marks Standard +0.8

The weights of eggs, $E$ grams, follow a normal distribution, $\mathrm { N } \left( 60,3 ^ { 2 } \right)$

The weights of empty small boxes, $S$ grams, follow a normal distribution, $\mathrm { N } \left( 24,1.8 ^ { 2 } \right)$
The weights of empty large boxes, $L$ grams, follow a normal distribution, $\mathrm { N } \left( 40,2.1 ^ { 2 } \right)$
Small boxes of eggs contain 6 randomly selected eggs.
Large boxes of eggs contain 12 randomly selected eggs.

Find the probability that the total weight of a randomly selected small box of 6 eggs weighs less than 387 grams.
Find the probability that a randomly selected large box of 12 eggs weighs more than twice a randomly selected small box of 6 eggs.

Edexcel FS2 2023 June Q5

9 marks Standard +0.3

A psychologist claims to have developed a technique to improve a person's memory.

A random sample of 8 people are each given the same list of words to memorise and recall. Each person then receives memory training from the psychologist. After the training, each person is given the same list of new words to memorise and recall. The table shows the percentage of words recalled by each person before and after the training.

Person	A	$B$	C	$D$	E	$F$	G	$H$
Percentage of words recalled before training	24	33	33	39	30	38	32	34
Percentage of words recalled after training	28	30	37	41	32	44	35	34

State why a paired $t$-test is suitable for these data.
State an assumption that needs to be made in order to carry out a paired $t$-test in this case.
Test, at the $5 \%$ level of significance, whether or not there is evidence of an increase in the percentage of words recalled after receiving the psychologist's training. State your hypotheses, test statistic and critical value used for this test.

Edexcel FS2 2023 June Q6

10 marks Challenging +1.2

The continuous random variable $X$ has cumulative distribution function given by

$$F ( x ) = \left\{ \begin{array} { c r } 0 & x < 0 \\ k \left( x - a x ^ { 2 } \right) & 0 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$ The values of $a$ and $k$ are positive constants such that $\mathrm { P } ( X < 2 ) = \frac { 2 } { 3 }$

Find the exact value of the median of $X$
Find the probability density function of $X$
Hence, deduce the value of the mode of $X$, giving a reason for your answer.

Edexcel FS2 2023 June Q7

9 marks Standard +0.3

The random variable $R$ has a continuous uniform distribution over the interval $[ 2,10 ]$
1. Write down the probability density function $\mathrm { f } ( r )$ of $R$
A sphere of radius $R \mathrm {~cm}$ is formed.
The surface area of the sphere, $S \mathrm {~cm} ^ { 2 }$, is given by $S = 4 \pi R ^ { 2 }$
Show that $\mathrm { E } ( S ) = \frac { 496 \pi } { 3 }$ The volume of the sphere, $V \mathrm {~cm} ^ { 3 }$, is given by $V = \frac { 4 } { 3 } \pi R ^ { 3 }$
Find, using algebraic integration, the expected value of $V$

Edexcel FS2 2023 June Q8

12 marks Challenging +1.2

A bag contains a large number of marbles of which an unknown proportion, $p$, is yellow.

Three random samples of size $n$ are taken, and the number of yellow marbles in each sample, $Y _ { 1 } , Y _ { 2 }$ and $Y _ { 3 }$, is recorded. Two estimators $\hat { \mathrm { p } } _ { 1 }$ and $\hat { \mathrm { p } } _ { 2 }$ are proposed to estimate the value of $p$ $$\begin{aligned} & \hat { p } _ { 1 } = \frac { Y _ { 1 } + 3 Y _ { 2 } - 2 Y _ { 3 } } { 2 n } \\ & \hat { p } _ { 2 } = \frac { 2 Y _ { 1 } + 3 Y _ { 2 } + Y _ { 3 } } { 6 n } \end{aligned}$$

Show that $\hat { \mathrm { p } } _ { 1 }$ and $\hat { \mathrm { p } } _ { 2 }$ are both unbiased estimators of $p$
Find the variance of $\hat { p } _ { 1 }$ The variance of $\hat { \mathrm { p } } _ { 2 }$ is $\frac { 7 p ( 1 - p ) } { 18 n }$
State, giving a reason, which is the better estimator. The estimator $\hat { p } _ { 3 } = \frac { Y _ { 1 } + a Y _ { 2 } + 3 Y _ { 3 } } { b n }$ where $a$ and $b$ are positive integers.
Find the pair of values of $a$ and $b$ such that $\hat { \mathrm { p } } _ { 3 }$ is a better unbiased estimator of $p$ than both $\hat { \mathrm { p } } _ { 1 }$ and $\hat { \mathrm { p } } _ { 2 }$
You must show all stages of your working.

Edexcel FS2 2024 June Q1

9 marks Standard +0.3

Two students are experimenting with some water in a plastic bottle. The bottle is filled with water and a hole is put in the bottom of the bottle. The students record the time, $t$ seconds, it takes for the water level to fall to each of 10 given values of the height, $h \mathrm {~cm}$, above the hole.

Student $A$ models the data with an equation of the form $t = a + b \sqrt { h }$
The data is coded using $v = t - 40$ and $w = \sqrt { h }$ and the following information is obtained. $$\sum v = 626 \quad \sum v ^ { 2 } = 64678 \quad \sum w = 22.47 \quad \mathrm {~S} _ { w w } = 4.52 \quad \mathrm {~S} _ { v w } = - 338.83$$

Find the equation of the regression line of $t$ on $\sqrt { h }$ in the form $t = a + b \sqrt { h }$ The time it takes the water level to fall to a height of 9 cm above the hole is 47 seconds.
Calculate the residual for this data point. Give your answer to 2 decimal places. Given that the residual sum of squares (RSS) for the model of $t$ on $\sqrt { h }$ is the same as the RSS for the model of $v$ on $w$,
calculate the RSS for these 10 data points. Student $B$ models the data with an equation of the form $t = c + d h$
The regression line of $t$ on $h$ is calculated and the residual sum of squares (RSS) is found to be 980 to 3 significant figures.
With reference to part (c) state, giving a reason, whether Student B's model or Student A's model is the more suitable for these data.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Sample size	\(\sum x\)	\(\sum x ^ { 2 }\)
Machine \(A\)	9	2268	571700

Patient	\(A\)	\(B\)	C	\(D\)	\(E\)	\(F\)	\(G\)
Resting heart rate before	65	68	77	79	80	88	92
Resting heart rate after	63	65	73	76	80	84	80

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	number of samples	\(S _ { A } ^ { 2 }\)
Site \(A\)	13	6.39

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