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AQA Further Paper 3 Statistics 2020 June Q4

9 marks Standard +0.3

4 The discrete random variable $X$ follows a discrete uniform distribution and takes values $1,2,3 , \ldots , n$. The discrete random variable $Y$ is defined by $Y = 2 X$ 4

Using the standard results for $\sum n , \sum n ^ { 2 }$ and $\operatorname { Var } ( a X + b )$, prove that $$\operatorname { Var } ( Y ) = \frac { n ^ { 2 } - 1 } { 3 }$$ 4
A spinning toy can land on one of four values: 2, 4, 6 or 8
Using a discrete uniform distribution, find the probability that the next value the toy lands on is greater than 2 4
State an assumption that is required for the discrete uniform distribution used in part (b) to be valid.

AQA Further Paper 3 Statistics 2020 June Q5

7 marks Challenging +1.2

5 Emily claims that the average number of runners per minute passing a shop during a long distance run is 8 Emily conducts a hypothesis test to investigate her claim.
5

State the hypotheses for Emily's test. 5
Emily counts the number of runners, $X$, passing the shop in a randomly chosen minute. The critical region for Emily's test is $X \leq 2$ or $X \geq 14$ During a randomly chosen minute, Emily counts 3 runners passing the shop.
Determine the outcome of Emily's hypothesis test.
5
The actual average number of runners per minute passing the shop is 7 Find the power of Emily's hypothesis test, giving your answer to three significant figures.

AQA Further Paper 3 Statistics 2020 June Q6

8 marks Standard +0.3

6 The distance, $X$ metres, between successive breaks in a water pipe is modelled by an exponential distribution. The mean of $X$ is 25 The distance between two successive breaks is measured. A water pipe is given a 'Red' rating if the distance is less than $d$ metres. The government has introduced a new law changing $d$ to 2
Before the government introduced the new law, the probability that a water pipe is given a 'Red' rating was 0.05 6

Explain whether or not the probability that a water pipe is given a 'Red' rating has increased as a result of the new law.
6
Find the probability density function of the random variable $X$. 6
After investigation, the distances between successive breaks in water pipes are found to have a standard deviation of 5 metres. Explain whether or not the use of an exponential model in parts (a) and (b) is appropriate.
[0pt] [2 marks]

AQA Further Paper 3 Statistics 2020 June Q7

8 marks Standard +0.3

7 The rainfall per day in February in a particular town has been recorded as having a mean of 1.8 inches. Sienna claims that rainfall in February has increased in the town. She records the rainfall in a random sample of 12 days. Her sample mean is 2 inches and her sample standard deviation is 0.4 inches.
It is assumed that rainfall per day has a normal distribution.
7

Investigate Sienna's claim using the $5 \%$ level of significance.
7
For the test carried out in part (a), state in context the meaning of a Type II error. 7
The distribution of rainfall per day in February in the town over 10 years is shown in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{443e7f17-a555-41ff-9d91-541cf45aae99-11_508_645_849_699} Explain whether or not the assumption that rainfall per day in February has a normal distribution is appropriate.

AQA Further Paper 3 Statistics 2020 June Q8

6 marks Standard +0.3

8 Ray is conducting a hypothesis test with the hypotheses $\mathrm { H } _ { 0 }$ : There is no association between time of day and number of snacks eaten $\mathrm { H } _ { 1 }$ : There is an association between time of day and number of snacks eaten
He calculates expected frequencies correct to two decimal places, which are given in the following table.

		Number of snacks eaten
\cline { 2 - 5 }\cline { 2 - 4 }		0	1	2 or more
\cline { 2 - 4 } Time of Day	23.68	21.05	5.26
\cline { 2 - 5 }	Night	21.32	18.95	4.74
\cline { 2 - 5 }
\cline { 2 - 5 }

Ray calculates his test statistic using $\sum \frac { ( O - E ) ^ { 2 } } { E }$ 8

State, with a reason, the error Ray has made and describe any changes Ray will need to make to his test.
8
Having made the necessary corrections as described in part (a), the correct value of the test statistic is 8.74 Complete Ray's hypothesis test using a $1 \%$ level of significance.

AQA Further Paper 3 Statistics 2020 June Q9

6 marks Challenging +1.2

9 The continuous random variable $X$ has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2 \\ 1 & x > 2 \end{array} \right.$$ The discrete random variable $Y$ has the probability distribution shown below.

$y$	2	7	13	19
$\mathrm { P } ( Y = y )$	0.5	0.1	0.1	0.3

The random variables $X$ and $Y$ are independent.
Find the exact value of $\mathrm { E } \left( X ^ { 3 } + Y \right)$.

Find the mean points scored when the game is played once, giving your answer to two decimal places.
3
Mina plays the game.
Her father, Michael, tells her that he will multiply her score by 5 and then subtract 10 He will then give her the value he has calculated in pence rounded to the nearest penny. Calculate the expected value in pence that Mina receives.

AQA Further Paper 3 Statistics 2021 June Q4

6 marks Standard +0.3

4 Oscar is studying the daily maximum temperature in ${ } ^ { \circ } \mathrm { C }$ in a village during the month of June. He constructs a $95 \%$ confidence interval of width $0.8 ^ { \circ } \mathrm { C }$ using a random sample of 150 days. He assumes that the daily maximum temperature has a normal distribution.
4

Find the standard deviation of Oscar's sample, giving your answer to three significant figures.
4
Oscar calculates the mean of his sample to be $25.3 ^ { \circ } \mathrm { C }$ He claims that the population mean is $26.0 ^ { \circ } \mathrm { C }$ Explain whether or not his confidence interval supports his claim.
4
Explain how Oscar could reduce the width of his 95\% confidence interval.

AQA Further Paper 3 Statistics 2021 June Q5

6 marks Standard +0.3

5 The continuous random variable $X$ has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x \leq 1 \\ \frac { 1 } { 10 } x - \frac { 1 } { 10 } & 1 < x \leq 6 \\ \frac { 1 } { 90 } x ^ { 2 } + \frac { 1 } { 10 } & 6 < x \leq 9 \\ 1 & x > 9 \end{array} \right.$$ 5

Find the probability density function $\mathrm { f } ( x )$ 5
Show that $\operatorname { Var } ( X ) = \frac { 6737 } { 1200 }$ \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-07_2488_1716_219_153}

AQA Further Paper 3 Statistics 2021 June Q6

7 marks Moderate -0.5

6 Danai is investigating the number of speeding offences in different towns in a country. She carries out a hypothesis test to test for association between town and number of speeding offences per year. 6

State the hypotheses for this test. 6
The observed frequencies, $O$, have been collected and the expected frequencies, $E$, have been calculated in an $n \times m$ contingency table, where $n > 3$ and $m > 3$ One of the values of $E$ is less than 5 6
1. Explain what steps Danai should take before calculating the test statistic.
  6
(ii) State an expression for the test statistic Danai should calculate.
6
Danai correctly calculates the value of the test statistic to be 45.22 The number of degrees of freedom for the test is 25
Determine the outcome of Danai's test, using the $1 \%$ level of significance.

AQA Further Paper 3 Statistics 2021 June Q7

11 marks Standard +0.3

7 The random variable $X$ has an exponential distribution with parameter $\lambda$ 7

Prove that $\mathrm { E } ( X ) = \frac { 1 } { \lambda }$ 7
Prove that $\operatorname { Var } ( X ) = \frac { 1 } { \lambda ^ { 2 } }$

AQA Further Paper 3 Statistics 2021 June Q8

13 marks Challenging +1.2

8 A company records the number of complaints, $X$, that it receives over 60 months. The summarised results are $$\sum x = 102 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 103.25$$ 8

Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7
8
The company also receives enquiries as well as complaints. The number of enquiries received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product.
The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the $10 \%$ level of significance.
8
It is later found that the mean total number of complaints and enquiries received per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places. \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-15_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-20_2496_1723_214_148}

AQA Further Paper 3 Statistics 2022 June Q1

1 marks Easy -1.2

1 The random variable $T$ follows a discrete uniform distribution and can take values $1,2,3 , \ldots , 16$ Find the variance of $T$ Circle your answer.
1.2518 .7521 .2521 .33

AQA Further Paper 3 Statistics 2022 June Q2

1 marks Moderate -0.8

2 The random variable $X$ has probability density function $$f ( x ) = \begin{cases} 1 & 0 < x \leq \frac { 1 } { 2 } \\ \frac { 3 } { 8 } x ^ { - 2 } & \frac { 1 } { 2 } < x \leq \frac { 3 } { 2 } \\ 0 & \text { otherwise } \end{cases}$$ Find $\mathrm { P } ( X < 1 )$ Circle your answer.
[0pt] [1 mark] $\frac { 1 } { 8 }$ $\frac { 3 } { 8 }$ $\frac { 5 } { 8 }$ $\frac { 7 } { 8 }$ \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-03_2488_1718_219_153}

AQA Further Paper 3 Statistics 2022 June Q3

5 marks Moderate -0.8

3 The random variable $X$ has an exponential distribution with probability density function $\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }$ where $x \geq 0$ 3

Show that the cumulative distribution function, for $x \geq 0$, is given by $\mathrm { F } ( x ) = 1 - \mathrm { e } ^ { - \lambda x }$ [0pt] [3 marks]
3
Given that $\lambda = 2$, find $\mathrm { P } ( X > 1 )$, giving your answer to three decimal places.

AQA Further Paper 3 Statistics 2022 June Q4

5 marks Standard +0.3

4 Daisies and dandelions are the only flowers growing in a field. The number of daisies per square metre in the field has a mean of 16
The number of dandelions per square metre in the field has a mean of 10
The number of daisies per square metre and the number of dandelions per square metre are independent. 4

Using a Poisson model, find the probability that a randomly selected square metre from the field has a total of at least 30 flowers, giving your answer to three decimal places.
4
A survey of the entire field is taken.
The standard deviation of the total number of flowers per square metre is 10 State, with a reason, whether the model used in part (a) is valid.

AQA Further Paper 3 Statistics 2022 June Q5

6 marks Standard +0.3

5 The mass, $X$, in grams of a particular type of apple is modelled using a normal distribution. A random sample of 12 apples is collected and the summarised results are $$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$ 5

A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample. Show that the confidence interval is $( 81.7,91.3 )$ with values correct to three significant figures.
5
Padraig claims that the population mean mass of the apples is 85 grams. He carries out a hypothesis test at the $1 \%$ level of significance using the random sample of 12 apples. The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 85 \\ & \mathrm { H } _ { 1 } : \mu \neq 85 \end{aligned}$$ State, with a reason, whether the null hypothesis is accepted or rejected.
5
Interpret, in context, the conclusion to the hypothesis test in part (b).

AQA Further Paper 3 Statistics 2022 June Q6

8 marks Standard +0.3

6 The discrete random variable $X$ has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 0 \\ b & x = 1 \\ c & x = 2 \\ 0 & \text { otherwise } \end{cases}$$ where $a , b$ and $c$ are constants.
The mean of $X$ is 1.2 and the variance of $X$ is 0.56
6

Deduce the values of $a , b$ and $c$ 6
The continuous random variable $Y$ is independent of $X$ and has variance 15 Find $\operatorname { Var } ( X - 2 Y - 11 )$ [0pt] [2 marks]

AQA Further Paper 3 Statistics 2022 June Q7

9 marks Easy -1.2

Test the scientist's claim, using the 10\% level of significance.
7
For the context of the test carried out in part (a), state the meaning of a Type I error. [1 mark]

AQA Further Paper 3 Statistics 2022 June Q8

11 marks Standard +0.3

8 The continuous random variable $X$ has cumulative distribution function $\mathrm { F } ( x )$ where $$\mathrm { F } ( x ) = \begin{cases} 0 & x = 0 \\ \mathrm { e } ^ { k x } - 1 & 0 \leq x \leq 5 \\ 1 & x > 5 \end{cases}$$ 8

Show that $k = \frac { 1 } { 5 } \ln 2$ [0pt] [2 marks]
8
Show that the median of $X$ is $a \frac { \ln b } { \ln 2 } - c$, where $a , b$ and $c$ are integers to be found.

8
Show that the mean of $X$ is $p - \frac { q } { \ln 2 }$, where $p$ and $q$ are integers to be found.

AQA Further Paper 3 Statistics 2022 June Q9

4 marks Moderate -0.8

9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12 9

The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to $p$ Lianne will only buy the battery if $p$ is more than 0.4
Determine whether Lianne will buy the battery.
[0pt] [2 marks]
9
A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427} Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
\includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}