Questions - Edexcel FS2 - A-Level Maths

find a $95 \%$ confidence interval for the mean weight, $\mu$ grams, of yoghurt in a pot. Give your answers to 2 decimal places.
Comment on whether or not the machine is working properly, giving a reason for your answer.
State the probability that a $95 \%$ confidence interval for $\mu$ will not contain $\mu$ grams.
Without carrying out any further calculations, explain the changes, if any, that would need to be made in calculating the confidence interval in part (a) if the standard deviation was unknown. Give a reason for your answer.
You may assume that the weights of the yoghurt delivered by the machine still follow a normal distribution.

Edexcel FS2 2019 June Q2

2 A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, $f$ grams $/ \mathrm { m } ^ { 2 }$. The yield of wheat, $w$ tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w ^ { 2 } = 13447 \quad \mathrm {~S} _ { f f } = 42 \quad \mathrm {~S} _ { f w } = 269.5$$

Calculate the product moment correlation coefficient between $f$ and $w$
Interpret the value of your product moment correlation coefficient.
Find the equation of the regression line of $w$ on $f$ in the form $w = a + b f$
Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams $/ \mathrm { m } ^ { 2 }$ The residuals of the data recorded are calculated and plotted on the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{67df73d4-6ce4-45f7-8a69-aa94292ea814-04_1232_1294_1169_301}
With reference to this graph, comment on the suitability of the model you found in part (c).
Suggest how you might be able to refine your model.

Edexcel FS2 2019 June Q3

3 Yin grows two varieties of potato, plant $A$ and plant $B$. A random sample of each variety of potato is taken and the yield, $x \mathrm {~kg}$, produced by each plant is measured. The following statistics are obtained from the data.

	Number of plants	$\sum x$	$\sum x ^ { 2 }$
$A$	25	194.7	1637.37
$B$	26	227.5	2031.19

Stating your hypotheses clearly, test, at the $10 \%$ significance level, whether or not the variances of the yields of the two varieties of potato are the same.
State an assumption you have made in order to carry out the test in part (a).

Edexcel FS2 2019 June Q4

4 The continuous random variable $X$ has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 0
k \left( x ^ { 3 } - \frac { 3 } { 8 } x ^ { 4 } \right) & 0 < x \leqslant 2
1 & x > 2 \end{array} \right.$$ where $k$ is a constant.

Show that $k = \frac { 1 } { 2 }$
Showing your working clearly, use calculus to find
1. $\mathrm { E } ( X )$
2. the mode of $X$
Describe, giving a reason, the skewness of the distribution of $X$

Edexcel FS2 2019 June Q5

5 Alexa believes that students are equally likely to achieve the same percentage score on each of two tests, paper I and paper II. She randomly selects 8 students and gives them each paper I and paper II. The percentage scores for each paper are recorded. The following paired data are collected.

Student	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Paper I (\%)	70	70	84	80	64	65	65	90
Paper II (\%)	64	76	72	74	68	64	58	76

Test, at the $1 \%$ significance level, whether or not there is evidence to support Alexa's belief. State your hypotheses clearly and show your working.

Edexcel FS2 2019 June Q6

6 A company manufactures bolts. The diameter of the bolts follows a normal distribution with a mean diameter of 5 mm . Stan believes that the mean diameter of the bolts is less than 5 mm . He takes a random sample of 10 bolts and measures their diameters. He calculates some statistics but spills ink on his work before completing them. The only information he has left is as follows
\includegraphics[max width=\textwidth, alt={}, center]{67df73d4-6ce4-45f7-8a69-aa94292ea814-16_394_1150_527_456} Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not Stan's belief is supported.

Edexcel FS2 2019 June Q7

7 A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic. The weights, $W \mathrm {~kg}$, of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg . The weights, $X \mathrm {~kg}$, of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg . The random variables $W$ and $X$ are independent.

Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy. The manufacturer packs $n$ of these wooden toys and $2 n$ of these plastic toys into the same container. The maximum weight the container can hold is 252 kg . The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.
Calculate the value of $n$.

Edexcel FS2 2019 June Q8

8 Nine athletes, $A , B , C , D , E , F , G , H$ and $I$, competed in both the 100 m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman's rank correlation coefficient was calculated and found to be 0.85

Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher an athlete's position is in the 100 m sprint, the higher their position is in the long jump. Use a $5 \%$ level of significance. The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes $B , C$ and $D$ over their long jump results. The table shows the results that are agreed to be correct.
Athlete $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$
Position in 100 m sprint 4 6 7 9 2 8 3 1 5
Position in long jump 5 4 9 3 1 2
Given that there were no tied ranks,
find the correct positions of athletes $B , C$ and $D$ in the long jump. You must show your working clearly and give reasons for your answers.
Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete $H$ was disqualified from both the 100 m sprint and the long jump.

Edexcel FS2 2020 June Q1

1 Gina receives a large number of packages from two companies, $A$ and $B$. She believes that the variance of the weights of packages from company $A$ is greater than the variance of the weights of packages from company $B$. Gina takes a random sample of 7 packages from company $A$ and an independent random sample of 10 packages from company $B$. Her results are summarised below $$\bar { a } = 300 \quad \mathrm {~S} _ { a a } = 145496 \quad \bar { b } = 233.4 \quad \mathrm {~S} _ { b b } = 56364.4$$ [You may assume that the weights of packages from the two companies are normally distributed.]
Test Gina's belief. Use a $5 \%$ level of significance and state your hypotheses clearly.

Edexcel FS2 2020 June Q2

2 Jemima makes jam to sell in a local shop. The jam is sold in jars and the weight of jam in a jar is normally distributed. Jemima takes a random sample of 8 of her jars of jam and weighs the contents of each jar, $x$ grams. Her results are summarised as follows $$\sum x = 3552 \quad \sum x ^ { 2 } = 1577314$$

Calculate a 95\% confidence interval for the mean weight of jam in a jar. The labels on the jars state that the average contents weigh 440 grams.
State, giving a reason, whether or not Jemima should be concerned about the labels on her jars of jam.

Edexcel FS2 2020 June Q3

3 Below are 3 sketches from some students of the residuals from their linear regressions of $y$ on $x$.
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_252_704_342_660}
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_266_718_625_660}
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_248_599_936_660} \section*{III} III For each sketch you should state, giving your reason,

whether or not the sketch is feasible
and if it is feasible
whether or not the sketch suggests a linear or a non-linear relationship between $y$ and $x$.

Edexcel FS2 2020 June Q4

4 A biased coin has a probability $p$ of landing on heads, where $0 < p < 1$ Simon spins the coin $n$ times and the random variable $X$ represents the number of heads. Taruni spins the coin $m$ times, $m \neq n$, and the random variable $Y$ represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of $p$.
Simon proposes the estimator $S = \frac { X + Y } { m + n }$ and Taruni proposes $T = \frac { 1 } { 2 } \left[ \frac { X } { n } + \frac { Y } { m } \right]$

Show that both $S$ and $T$ are unbiased estimators of $p$.
Prove that, for all values of $m$ and $n , S$ is the better estimator.

Edexcel FS2 2020 June Q5

5 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{54bf68ab-7934-432a-890f-20093082ab07-12_446_1105_242_479} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The random variable $X$ has probability density function $\mathrm { f } ( x )$ and Figure 1 shows a sketch of $\mathrm { f } ( x )$ where $$f ( x ) = \left\{ \begin{array} { c c } k ( 1 - \cos x ) & 0 \leqslant x \leqslant 2 \pi
0 & \text { otherwise } \end{array} \right.$$

Show that $k = \frac { 1 } { 2 \pi }$ The random variable $Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$ and $\mathrm { E } ( Y ) = \mathrm { E } ( X )$
The probability density function of $Y$ is $g ( y )$, where $$g ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } e ^ { - \frac { 1 } { 2 } \left( \frac { y - \mu } { \sigma } \right) ^ { 2 } } \quad - \infty < y < \infty$$ Given that $\mathrm { g } ( \mu ) = \mathrm { f } ( \mu )$
find the exact value of $\sigma$
Calculate the error in using $\mathrm { P } \left( \frac { \pi } { 2 } < Y < \frac { 3 \pi } { 2 } \right)$ as an approximation to $\mathrm { P } \left( \frac { \pi } { 2 } < X < \frac { 3 \pi } { 2 } \right)$

Edexcel FS2 2020 June Q6

6 A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods.
Each day a random sample of rods is taken, their lengths measured and a $95 \%$ confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed. Kim took a random sample of 25 rods and used the $t$ distribution to obtain a $95 \%$ confidence interval of $( 1.193,1.367 )$ for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres.

Test, at the $10 \%$ level of significance, whether or not Kim's sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly. Using Kim's sample and the normal distribution with a standard deviation of 0.175 metres, (b) find a 95\% confidence interval for the mean length of the rods.

Edexcel FS2 2020 June Q7

7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm . The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm .

Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. One large panel and one small panel are selected at random.
Find the probability that the length of the large panel is more than $\frac { 8 } { 3 }$ times the length of the small panel. Rosa needs 1000 cm of fencing. The large panels cost $\pounds 80$ each and the small panels cost $\pounds 30$ each. Rosa's plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.
Calculate whether or not the expected cost of Rosa's plan is cheaper than simply buying 14 small panels.

Edexcel FS2 2020 June Q8

8 A circle, centre $O$, has radius $x \mathrm {~cm}$, where $x$ is an observation from the random variable $X$ which has a rectangular distribution on $[ 0 , \pi ]$

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

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

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

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

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

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

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

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

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

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.

Athlete	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)
Position in 100 m sprint	4	6	7	9	2	8	3	1	5
Position in long jump	5				4	9	3	1	2

	Number of plants	\(\sum x\)	\(\sum x ^ { 2 }\)
\(A\)	25	194.7	1637.37
\(B\)	26	227.5	2031.19

Questions — Edexcel FS2 (54 questions)

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