Questions S2 - A-Level Maths

Edexcel S2 Q3

9 marks Standard +0.3

3. A primary school teacher finds that exactly half of his year 6 class have mobile phones. He decides to investigate whether the proportion of pupils with mobile phones is different from 0.5 in the year 5 class at his school. There are 25 pupils in the year 5 class.

State the hypotheses that he should use.
Find the largest critical region for this test such that the probability in each "tail" is less than $2.5 \%$.
Determine the significance level of this test. He finds that eight of the year 5 pupils have mobile phones and concludes that there is not sufficient evidence of the proportion being different from 0.5
Stating the new hypotheses clearly, find if the number of year 5 pupils with mobile phones would have been significant if he had tested whether or not the proportion was less than 0.5 and used the largest critical region with a probability of less than $5 \%$.
(3 marks)

Edexcel S2 Q4

10 marks Standard +0.3

4. A hardware store is open on six days each week. On average the store sells 8 of a particular make of electric drill each week. Find the probability that the store sells

no more than 4 of the drills in a week,
more than 2 of the drills in one day. The store receives one delivery of drills at the same time each week.
Find the number of drills that need to be in stock after a delivery for there to be at most a 5\% chance of the store not having sufficient drills to meet demand before the next delivery.
(3 marks)
[0pt]

Edexcel S2 Q5

10 marks Moderate -0.3

5. In a party game, a bottle is spun and whoever it points to when it stops has to play next. The acute angle, in degrees, that the bottle makes with the side of the room is modelled by a rectangular distribution over the interval [0,90]. Find the probability that on one spin this angle is

between $25 ^ { \circ }$ and $38 ^ { \circ }$,
$45 ^ { \circ }$ to the nearest degree. The bottle is spun ten times.
Find the probability that the acute angle it makes with the side of the room is less than $10 ^ { \circ }$ more than twice.

Edexcel S2 Q6

12 marks Standard +0.3

6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.

Find the significance level of this test.
State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
Using a suitable approximation, test at the $5 \%$ level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
(6 marks)

Edexcel S2 Q7

18 marks Standard +0.3

7. In a competition at a funfair, participants have to stay on a log being rotated in a pool of water for as long as possible. The length of time, in tens of seconds, that the competitors stay on the log is modelled by the random variable $T$ with the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k ( t - 3 ) ^ { 2 } , & 0 \leq t \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$

Show that $k = \frac { 1 } { 9 }$.
Sketch f $( t )$ for all values of $t$.
Show that the mean time that competitors stay on the $\log$ is 7.5 seconds. When the competition is next run the organisers decide to make it easier at first by spinning the log more slowly and then increasing the speed of rotation. The length of time, in tens of seconds, that the competitors now stay on the log is modelled by the random variable $S$ with the following probability density function: $$f ( s ) = \begin{cases} \frac { 1 } { 12 } \left( 8 - s ^ { 3 } \right) , & 0 \leq s \leq 2 \\ 0 , & \text { otherwise } \end{cases}$$
Find the change in the mean time that competitors stay on the log.

Edexcel S2 Q1

7 marks Easy -1.2

The random variable $X$ follows a Poisson distribution with a mean of 1.4 Find $\mathrm { P } ( X \leq 3 )$.
The random variable $Y$ follows a binomial distribution such that $Y \sim \mathrm {~B} ( 20,0.6 )$. Find $\mathrm { P } ( Y \leq 12 )$.
(4 marks)

Edexcel S2 Q2

9 marks Moderate -0.8

2. A driving instructor keeps records of all the learners she has taught. In order to analyse her success rate she wishes to take a random sample of 120 of these learners.

Suggest a suitable sampling frame and identify the sampling units. She believes that only 1 in 20 of the people she teaches fail to pass their test in their first two attempts. She decides to use her sample to test whether or not the proportion is different from this.
Using a suitable approximation and stating clearly the hypotheses she should use, find the largest critical region for this test such that the probability in each "tail" is less than $2.5 \%$.
State the significance level of this test.

Edexcel S2 Q3

9 marks Standard +0.3

3. In an old computer game a white square representing a ball appears at random at the top of the playing area, which is 24 cm wide, and moves down the screen. The continuous random variable $X$ represents the distance, in centimetres, of the dot from the left-hand edge of the screen when it appears. The distribution of $X$ is rectangular over the interval [4,28].

Find the mean and variance of $X$.
Find $\mathrm { P } ( | X - 16 | < 3 )$. During a single game, a player receives 12 "balls".
Find the probability that the ball appears within 3 cm of the middle of the top edge of the playing area more than four times in a single game.
(3 marks)

Edexcel S2 Q4

14 marks Moderate -0.8

4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.

State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case. Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
less than two visitors in a 10 -minute interval,
at least ten visitors in a 15-minute interval.
Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.
(5 marks)

Edexcel S2 Q5

17 marks Standard +0.3

5. Four coins are flipped together and the random variable $H$ represents the number of heads obtained. Assuming that the coins are fair,

suggest with reasons a suitable distribution for modelling $H$ and give the value of any parameters needed,
show that the probability of obtaining more heads than tails is $\frac { 5 } { 16 }$. The four coins are flipped 5 times and more heads are obtained than tails 4 times.
Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence of the probability of getting more heads than tails being more than $\frac { 5 } { 16 }$. Given that the four coins are all biased such that the chance of each one showing a head is 50\% more than the chance of it showing a tail,
find the probability of obtaining more heads than tails when the four coins are flipped together.

Edexcel S2 Q6

19 marks Standard +0.3

6. The continuous random variable $X$ has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 16 } x , & 2 \leq x \leq 6 \\ 0 , & \text { otherwise } \end{cases}$$

Sketch $\mathrm { f } ( x )$ for all values of $x$.
Find $\mathrm { E } ( X )$.
Show that $\operatorname { Var } ( X ) = \frac { 11 } { 9 }$.
Define fully the cumulative distribution function $\mathrm { F } ( x )$ of $X$.
Show that the interquartile range of $X$ is $2 ( \sqrt { } 7 - \sqrt { 3 } )$. END

CAIE S2 2019 June Q6

9 marks Standard +0.3

Show that $b = \frac { a } { a - 1 }$.
Given that the median of $X$ is $\frac { 3 } { 2 }$, find the values of $a$ and $b$.
Use your values of $a$ and $b$ from part (ii) to find $\mathrm { E } ( X )$.

AQA S2 2011 June Q3

10 marks Standard +0.3

State the null hypothesis that Emily used.
Find the value of the test statistic, $X ^ { 2 }$, giving your answer to one decimal place.
State, in context, the conclusion that Emily should reach based on the results of her $\chi ^ { 2 }$ test.
Make one comment on the GCSE performances of 16-year-old students attending Bailey Language School.
Emily's friend, Joanna, used the same data to correctly conduct a $\chi ^ { 2 }$ test using the $10 \%$ level of significance. State, with justification, the conclusion that Joanna should reach.

OCR S2 2007 June Q4

6 marks Moderate -0.3

State two conditions needed for $X$ to be well modelled by a normal distribution.
It is given that $X \sim \mathrm {~N} \left( 50.0,8 ^ { 2 } \right)$. The mean of 20 random observations of $X$ is denoted by $\bar { X }$. Find $\mathrm { P } ( \bar { X } > 47.0 )$. 5 The number of system failures per month in a large network is a random variable with the distribution $\operatorname { Po } ( \lambda )$. A significance test of the null hypothesis $\mathrm { H } _ { 0 } : \lambda = 2.5$ is carried out by counting $R$, the number of system failures in a period of 6 months. The result of the test is that $\mathrm { H } _ { 0 }$ is rejected if $R > 23$ but is not rejected if $R \leqslant 23$.
State the alternative hypothesis.
Find the significance level of the test.
Given that $\mathrm { P } ( R > 23 ) < 0.1$, use tables to find the largest possible actual value of $\lambda$. You should show the values of any relevant probabilities. 6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter $e$ appears $19 \%$ of the time. A certain encoded message of 20 letters contains one letter $e$.
Using an exact binomial distribution, test at the $10 \%$ significance level whether there is evidence that the proportion of the letter $e$ in the language from which this message is a sample is less than in German, i.e., less than $19 \%$.
Give a reason why a binomial distribution might not be an appropriate model in this context. 7 Two continuous random variables $S$ and $T$ have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
Sketch on the same axes the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$. [You should not use graph paper or attempt to plot points exactly.]
Explain in everyday terms the difference between the two random variables.
Find the value of $t$ such that $\mathrm { P } ( T > t ) = 0.2$. 8 A random variable $Y$ is normally distributed with mean $\mu$ and variance 12.25. Two statisticians carry out significance tests of the hypotheses $\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0$.
Statistician $A$ uses the mean $\bar { Y }$ of a sample of size 23, and the critical region for his test is $\bar { Y } > 64.20$. Find the significance level for $A$ 's test.
Statistician $B$ uses the mean of a sample of size 50 and a significance level of $5 \%$.
1. Find the critical region for $B$ 's test.
2. Given that $\mu = 65.0$, find the probability that $B$ 's test results in a Type II error.
3. Given that, when $\mu = 65.0$, the probability that $A$ 's test results in a Type II error is 0.1365 , state with a reason which test is better. 9 (a) The random variable $G$ has the distribution $\mathrm { B } ( n , 0.75 )$. Find the set of values of $n$ for which the distribution of $G$ can be well approximated by a normal distribution.
  (b) The random variable $H$ has the distribution $\mathrm { B } ( n , p )$. It is given that, using a normal approximation, $\mathrm { P } ( H \geqslant 71 ) = 0.0401$ and $\mathrm { P } ( H \leqslant 46 ) = 0.0122$.
  1. Find the mean and standard deviation of the approximating normal distribution.
  2. Hence find the values of $n$ and $p$.

AQA S2 2009 January Q1

11 marks Standard +0.3

1 Fortune High School gave its students a wider choice of subjects to study. The table shows the number of students, of each gender, who chose to study each of the additional subjects during the school year 2007/08.

\cline { 2 - 5 } \multicolumn{1}{c|}{}

Bulgarian

Climate

Change

Finance

Polish

Male

7

31

25

40

Female

2

24

22

19

Assuming that these data form a random sample, use a $\chi ^ { 2 }$ test, at the $10 \%$ level of significance, to test whether the choice of these subjects is independent of gender.
(11 marks)

AQA S2 2009 January Q2

9 marks Standard +0.3

2 A group of estate agents in a particular area claimed that, after the introduction of a new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area had not changed from 8 weeks.

A random sample of 9 house purchases in the area revealed that their completion times, in weeks, were as follows. $$\begin{array} { l l l l l l l l l } 6 & 7 & 10 & 12 & 9 & 11 & 7 & 8 & 14 \end{array}$$ Assuming that completion times in the area are normally distributed with standard deviation 2.5 weeks, test, at the $5 \%$ level of significance, the group's claim. (7 marks)
It was subsequently discovered that, after the introduction of the new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area remained at 8 weeks. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out your hypothesis test in part (a). Give a reason for your answer.
(2 marks)

AQA S2 2009 January Q3

14 marks Moderate -0.3

3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, $X$, for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, $Y$, for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.

Calculate:
1. $\mathrm { P } ( X \leqslant 3 )$;
2. $\quad \mathrm { P } ( Y = 5 )$.
The total daily demand for the device at Joe's two garages is denoted by $T$.
1. Write down the distribution of $T$, stating any assumption that you make.
2. Determine $\mathrm { P } ( 6 < T < 12 )$.
3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least $99 \%$ of days.

AQA S2 2009 January Q4

6 marks Moderate -0.3

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

Determine $\mathrm { P } \left( - \frac { 3 c } { 4 } < X < \frac { 3 c } { 4 } \right)$.
Show that the probability density function, $\mathrm { f } ( x )$, of $X$ is $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 c } & - c \leqslant x \leqslant 3 c \\ 0 & \text { otherwise } \end{array} \right.$$
Hence, or otherwise, find expressions, in terms of $c$, for:
1. $\mathrm { E } ( X )$;
2. $\operatorname { Var } ( X )$.

AQA S2 2009 January Q5

13 marks Standard +0.3

5 Jane, who supplies fruit to a jam manufacturer, knows that the weight of fruit in boxes that she sends to the manufacturer can be modelled by a normal distribution with unknown mean, $\mu$ grams, and unknown standard deviation, $\sigma$ grams. Jane selects a random sample of 16 boxes and, using the $t$-distribution, calculates correctly that a $98 \%$ confidence interval for $\mu$ is $( 70.65,80.35 )$.

1. Show that the sample mean is 75.5 grams.
2. Find the width of the confidence interval.
3. Calculate an estimate of the standard error of the mean.
4. Hence, or otherwise, show that an unbiased estimate of $\sigma ^ { 2 }$ is 55.6 , correct to three significant figures.
Jane decides that the width of the $98 \%$ confidence interval is too large. Construct a $95 \%$ confidence interval for $\mu$, based on her sample of 16 boxes.
Jane is informed that the manufacturer would prefer the confidence interval to have a width of at most 5 grams.
1. Write down a confidence interval for $\mu$, again based on Jane's sample of 16 boxes, which has a width of 5 grams.
2. Determine the percentage confidence level for your interval in part (c)(i).

AQA S2 2009 January Q6

10 marks Standard +0.3

6 A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for $R$, the number of checkouts that are staffed at any given time, is $$\mathrm { P } ( R = r ) = \left\{ \begin{array} { c l } \frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { r - 1 } & r = 1,2,3 \\ k & r = 4 \end{array} \right.$$

Show that $k = \frac { 1 } { 27 }$.
Find the probability that, at any given time, there will be at least 3 checkouts that are staffed.
It is suggested that the total number of customers, $C$, that can be served at the checkouts per hour may be modelled by $$C = 27 R + 5$$ Find:
1. $\mathrm { E } ( C )$;
2. the standard deviation of $C$.

AQA S2 2009 January Q7

12 marks Standard +0.3

7 The continuous random variable $X$ has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 16 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ \frac { 1 } { 6 } ( 5 - x ) & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$

Sketch the graph of f.
Prove that the cumulative distribution function of $X$ for $2 \leqslant x \leqslant 5$ can be written in the form $$\mathrm { F } ( x ) = 1 - \frac { 1 } { 12 } ( 5 - x ) ^ { 2 }$$
Hence, or otherwise, determine $\mathrm { P } ( X \geqslant 3 \mid X \leqslant 4 )$.

AQA S2 2007 June Q1

10 marks Standard +0.3

1 Two groups of patients, suffering from the same medical condition, took part in a clinical trial of a new drug. One of the groups was given the drug whilst the other group was given a placebo, a drug that has no physical effect on their medical condition. The table shows the number of patients in each group and whether or not their condition improved.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Placebo	Drug
Condition improved	20	46
Condition did not improve	55	29

Conduct a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to determine whether the condition of the patients at the conclusion of the trial is associated with the treatment that they were given.
(10 marks)

AQA S2 2007 June Q2

10 marks Moderate -0.8

2 The number of telephone calls per day, $X$, received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, $Y$, received by Candice may be modelled by a Poisson distribution with mean 6.0.

For any particular day, find:
1. $\mathrm { P } ( X = 3 )$;
2. $\quad \mathrm { P } ( Y \geqslant 5 )$.
1. Write down the distribution of $T$, the total number of telephone calls and e-mails per day received by Candice.
2. Determine $\mathrm { P } ( 7 \leqslant T \leqslant 10 )$.
3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
  (2 marks)

AQA S2 2007 June Q3

8 marks Standard +0.3

3 David is the professional coach at the golf club where Becki is a member. He claims that, after having a series of lessons with him, the mean number of putts that Becki takes per round of golf will reduce from her present mean of 36 . After having the series of lessons with David, Becki decides to investigate his claim.
She therefore records, for each of a random sample of 50 rounds of golf, the number of putts, $x$, that she takes to complete the round. Her results are summarised below, where $\bar { x }$ denotes the sample mean. $$\sum x = 1730 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 784$$ Using a $z$-test and the $1 \%$ level of significance, investigate David's claim.

AQA S2 2007 June Q4

7 marks Moderate -0.8

4 Students are each asked to measure the distance between two points to the nearest tenth of a metre.

Given that the rounding error, $X$ metres, in these measurements has a rectangular distribution, explain why its probability density function is $$f ( x ) = \left\{ \begin{array} { c c } 10 & - 0.05 < x \leqslant 0.05 \\ 0 & \text { otherwise } \end{array} \right.$$
Calculate $\mathrm { P } ( - 0.01 < X < 0.02 )$.
Find the mean and the standard deviation of $X$.

Questions S2 (1690 questions)

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Edexcel S2 Q3

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