Questions - OCR Further Statistics

OCR Further Statistics 2018 December Q7

12 marks Standard +0.8

7 Sasha tends to forget his passwords. He investigates whether the number of attempts he needs to log on to a system with a password can be modelled by a geometric distribution. On 60 occasions he records the number of attempts he needs to log on, and the results are shown in the table.

Number of attempts	1	2	3	4 or more
Frequency	20	19	13	3

Test at the $1 \%$ significance level whether the results are consistent with the distribution Geo(0.4).
[0pt]
Suggest which two probabilities should be changed, and in what way, to produce an improved model. (Numerical values are not required.) You should give a reason for your suggestion. [3]

OCR Further Statistics 2018 December Q8

11 marks Standard +0.8

8 A continuous random variable $X$ has probability density function given by the following function, where $a$ is a constant. $\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a , \\ 0 & \text { otherwise. } \end{array} \right\}$ The expected value of $X$ is 4 .

Show that $a = 6$. Five independent observations of $X$ are obtained, and the largest of them is denoted by $M$.
Find the cumulative distribution function of $M$. \section*{OCR} Oxford Cambridge and RSA

OCR Further Statistics 2021 June Q1

5 marks Standard +0.3

1 The performance of a piece of music is being recorded. The piece consists of three sections, $A , B$ and $C$. The times, in seconds, taken to perform the three sections are normally distributed random variables with the following means and standard deviations. \end{table}

Question

Answer

Mark

AO

Guidance

\multirow[t]{3}{*}{1}

\multirow[t]{3}{*}{(a)}

$A + B + C \sim \mathrm {~N} ( 701 , \ldots$

.. 419)

M1

1.1a

Normal, mean $\mu _ { A } + \mu _ { B } + \mu _ { C }$

\multirow{3}{*}{}

A1

1.1

Variance 419

$\mathrm { P } ( > 720 ) = 0.176649$

A1

1.1

Answer, 0.177 or better, www

\multirow[t]{2}{*}{1}

\multirow[t]{2}{*}{(b)}

$2 A + B \sim \mathrm {~N} ( 701,757 )$

M1

1.1a

Normal, same mean, $4 \sigma _ { A } { } ^ { 2 } + \sigma _ { B } { } ^ { 2 }$

\multirow{2}{*}{}

$\mathrm { P } ( > 720 ) = 0.244919$

A1 [2]

1.1

Answer, art 0.245

\multirow{2}{*}{2}

\multirow{2}{*}{(a)}

$\frac { { } ^ { 8 } C _ { 3 } \times { } ^ { 20 } C _ { 5 } } { { } ^ { 28 } C _ { 8 } }$

M1 A1

3.1b 1.1

(Product of two ${ } ^ { n } C _ { r }$ ) ÷ ${ } ^ { n } C _ { r }$ At least two ${ } ^ { n } C _ { r }$ correct

\multirow[t]{2}{*}{Or $\frac { 8 } { 28 } \times \frac { 7 } { 27 } \times \frac { 6 } { 26 } \times \frac { 20 } { 25 } \times \ldots \times \frac { 16 } { 21 } \times { } ^ { 8 } C _ { 3 } = 0.27934 \ldots$}

$\frac { 56 \times 15504 } { 3108105 } = 0.27934 \ldots$

A1 [3]

1.1

Any exact form or awrt 0.279

2

(b)

× B × B × B × B × B × B × B × B x

GGG in one $\mathrm { x } , \mathrm { G }$ in another: $9 \times 8$ $\div \frac { 12 ! } { 8 ! \times 4 ! }$ $= \frac { 72 } { 495 } = \frac { 8 } { 55 } \text { or } 0.145 \ldots$

M1 A1

3.1b 2.1

Or e.g. $\frac { 10 ! } { 8 ! } - 2 \times 9$

Divide by ${ } _ { 12 } \mathrm { C } _ { 4 }$ oe

Or, e.g. find ${ } _ { 12 } \mathrm { C } _ { 4 }$ - (\# (all separate) +\#(all together) $+ \# ( 2,1,1 ) \times 3 +$ \#(2,2))

M1

1.1

A1

1.1

[4]

Question

Answer

Mark

AO

Guidance

\multirow{7}{*}{3}

\multirow{7}{*}{(a)}

$\mathrm { H } _ { 0 } : \mu = 700$

B2

1.1

One error, e.g. no or wrong

Ignore failure to define $\mu$

$\mathrm { H } _ { 1 } : \mu < 700$ where $\mu$ is the mean reaction

1.1

letter, $\neq$, etc : B1

here

$\bar { x } = 607$

M1

3.3

Find sample mean

$z = - 1.822$ or $p = 0.0342$ or $\mathrm { CV } = 616.05 \ldots$

A1

3.4

Correct $z , p$ or CV

$z < - 1.645$ or $p < 0.05$ or $607 < \mathrm { CV }$

A1

1.1

Correct comparison

Reject $\mathrm { H } _ { 0 }$

M1ft

1.1

Correct first conclusion

Needs correct method, like-

Significant evidence that mean reaction times

A1ft

2.2b

Context, not too definite (e.g. not "international athletes' reaction times are shorter"

ft on their $z , p$ or CV

3

(b)

(i)

Uses more information (e.g. magnitudes of differences)

B1 [1]

2.4

\multirow{5}{*}{3}

\multirow{5}{*}{(b)}

\multirow{5}{*}{(ii)}

$\mathrm { H } _ { 0 } : m = 700 , \mathrm { H } _ { 1 } : m < 700$ where $m$ is the median reaction time for all international athletes

B1

2.5

Same as in (i) but different letter or "median" stated

$W _ { - } = 18$

$W _ { + } = 3$ so $T = 3$

For both, and $T$ correct

$n = 6 , \mathrm { CV } = 2$

A1

1.1

Correct CV

Do not reject $\mathrm { H } _ { 0 }$. Insufficient evidence that median reaction times of international athletes are shorter

A1ft [6]

2.2b

In context, not too definite

FT on their $T$

3

(c)

They use different assumptions

B1 [1]

2.3

Not "one is more accurate"

Question

Answer

Mark

AO

Guidance

4

(a)

\(\begin{aligned}

\int _ { 0 } ^ { a } x \frac { 2 x } { a ^ { 2 } } d x = 4

{ \left[ \frac { 2 x ^ { 3 } } { 3 a ^ { 2 } } \right] = 4 }

\frac { 2 } { 3 } a = 4 \Rightarrow a = 6 \end{aligned}\)

M1

B1

A1 [3]

3.1a

1.1

2.2a

4

(b)

$\mathrm { F } ( x ) = \frac { x ^ { 2 } } { 36 }$
Let the CDF of $M$ be $\mathrm { H } ( m )$. Then $\mathrm { H } ( m ) = \mathrm { P } ($ all observations less than $m )$ $= [ \mathrm { P } ( X \leqslant m ) ] ^ { 5 }$ $= \left[ \frac { m ^ { 2 } } { 36 } \right] ^ { 5 }$
\(\mathrm { H } ( m ) = \begin{cases} 0	m < 0 ,
\frac { m ^ { 10 } } { 60466176 }	0 \leq m \leq 6 ,
1	m > 6 . \end{cases}\)

M1 A1ft

M1

A1

[8]

1.1
1.1
2.1
3.1a
2.2a
2.1
2.1
1.2

Find $\mathrm { F } ( x ) ; = \frac { x ^ { 2 } } { a ^ { 2 } }$

Correct basis for CDF of $m$

Correct function, any letter Range $0 \leq m \leq 6$

Letter not $x$, and 0, 1 present

ft on their $a$

Allow

OCR Further Statistics 2021 June Q1

4 marks Moderate -0.8

1
The continuous random variable $X$ has the distribution $\mathrm { N } ( \mu , 30 )$. The mean of a random sample of 8 observations of $X$ is 53.1 . Determine a $95 \%$ confidence interval for $\mu$. You should give the end points of the interval correct to 4 significant figures.

OCR Further Statistics 2021 June Q2

12 marks Standard +0.3

2 A book collector compared the prices of some books, $\pounds x$, when new in 1972 and the prices of copies of the same books, $\pounds y$, on a second-hand website in 2018.
The results are shown in Table 1 and are summarised below the table. \begin{table}[h]

Book	A	B	C	D	E	F	G	H	I	J	K	L
$x$	0.95	0.65	0.70	0.90	0.55	1.40	1.50	0.50	1.15	0.35	0.20	0.35
$y$	6.06	7.00	2.00	5.87	4.00	5.36	7.19	2.50	3.00	8.29	1.37	2.00

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} $$n = 12 , \Sigma x = 9.20 , \Sigma y = 54.64 , \Sigma x ^ { 2 } = 8.9950 , \Sigma y ^ { 2 } = 310.4572 , \Sigma x y = 46.0545$$

It is given that the value of Pearson's product-moment correlation coefficient for the data is 0.381 , correct to 3 significant figures.
1. State what this information tells you about a scatter diagram illustrating the data.
2. Test at the $5 \%$ significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018.
The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J. Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books.

OCR Further Statistics 2021 June Q3

11 marks Standard +0.3

3 The numbers of CD players sold in a shop on three consecutive weekends were 7,6 and 2 . It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by $X$.

How appropriate is the Poisson distribution as a model for $X$ ? Now assume that a Poisson distribution with mean 5 is an appropriate model for $X$.
Find
1. $\mathrm { P } ( X = 6 )$,
2. $\mathrm { P } ( X \geqslant 8 )$. The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution $\operatorname { Po } ( 7.2 )$.
Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive.
State an assumption needed for your answer to part (c) to be valid.
Give a reason why the assumption in part (d) may not be valid in practice.

OCR Further Statistics 2021 June Q4

15 marks Standard +0.8

4 The continuous random variable $X$ has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where $n$ and $k$ are constants and $n$ is an integer greater than 1 .

Find $k$ in terms of $n$.
1. When $n = 4$, find the cumulative distribution function of $X$.
2. Hence determine $\mathrm { P } ( X > 7 \mid X > 5 )$ when $n = 4$.
Determine the values of $n$ for which $\operatorname { Var } ( X )$ is not defined.

OCR Further Statistics 2020 November Q1

4 marks Moderate -0.8

The continuous random variable $X$ has the distribution $\text{N}(\mu, 30)$. The mean of a random sample of 8 observations of $X$ is 53.1. Determine a 95\% confidence interval for $\mu$. You should give the end points of the interval correct to 4 significant figures. [4]

OCR Further Statistics 2020 November Q2

8 marks Standard +0.3

A book collector compared the prices of some books, $£x$, when new in 1972 and the prices of copies of the same books, $£y$, on a second-hand website in 2018. The results are shown in Table 1 and are summarised below the table.

Book	A	B	C	D	E	F	G	H	I	J	K	L
$x$	0.95	0.65	0.70	0.90	0.55	1.40	1.50	0.50	1.15	0.35	0.20	0.35
$y$	6.06	7.00	2.00	5.87	4.00	5.36	7.19	2.50	3.00	8.29	1.37	2.00

Table 1 $n = 12, \Sigma x = 9.20, \Sigma y = 54.64, \Sigma x^2 = 8.9950, \Sigma y^2 = 310.4572, \Sigma xy = 46.0545$

It is given that the value of Pearson's product-moment correlation coefficient for the data is 0.381, correct to 3 significant figures.
1. State what this information tells you about a scatter diagram illustrating the data. [1]
2. Test at the 5\% significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018. [5]
The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J. Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books. [2]

OCR Further Statistics 2020 November Q3

9 marks Challenging +1.2

Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B. She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219.

Test at the 10\% significance level whether route A has shorter journey times than route B. [8]
State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid. [1]

OCR Further Statistics 2020 November Q4

7 marks Standard +0.8

The random variable $X$ is equally likely to take any of the $n$ integer values from $m + 1$ to $m + n$ inclusive. It is given that $\text{E}(3X) = 30$ and $\text{Var}(3X) = 36$. Determine the value of $m$ and the value of $n$. [7]

OCR Further Statistics 2020 November Q5

11 marks Challenging +1.2

26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.

Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
All 26 cards are arranged in a line, in random order.
1. Show that the probability that all the vowels are next to one another is $\frac{1}{2990}$. [3]
2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]

OCR Further Statistics 2020 November Q6

11 marks Standard +0.3

The numbers of CD players sold in a shop on three consecutive weekends were 7, 6 and 2. It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by $X$.

How appropriate is the Poisson distribution as a model for $X$? [2]

Now assume that a Poisson distribution with mean 5 is an appropriate model for $X$.

Find
1. P$(X = 6)$, [2]
2. P$(X \geqslant 8)$. [2]

The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution Po(7.2).

Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive. [3]
State an assumption needed for your answer to part (c) to be valid. [1]
Give a reason why the assumption in part (d) may not be valid in practice. [1]

OCR Further Statistics 2020 November Q7

10 marks Standard +0.3

A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, $X$, up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency $f$ with which each value of $X$ is obtained. His results are shown in Table 1, together with the values of $xf$.

$x$	1	2	3	4	5	6	$\geqslant 7$	Total
Frequency $f$	20	15	9	13	10	10	23	100
$xf$	20	30	27	52	50	60	161	400

Table 1

State an appropriate distribution with which to model $X$, determining the value(s) of any parameter(s). [3]

Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places.

$x$	1	2	3	4	5	6	$\geqslant 7$
Observed frequency $O$	20	15	9	13	10	10	23
Expected frequency $E$	25	18.75	14.063	10.547	7.910	5.933	17.798
$(O - E)^2/E$	1	0.75	1.823	0.571	0.552	2.789	1.520

Table 2

Show how the expected frequency corresponding to $x \geqslant 7$ was obtained. [2]
Carry out the test. [5]

OCR Further Statistics 2020 November Q8

15 marks Standard +0.8

The continuous random variable $X$ has probability density function $$f(x) = \begin{cases} \frac{k}{x^n} & x \geqslant 1, \\ 0 & \text{otherwise}, \end{cases}$$ where $n$ and $k$ are constants and $n$ is an integer greater than 1.

Find $k$ in terms of $n$. [3]
1. When $n = 4$, find the cumulative distribution function of $X$. [3]
2. Hence determine P$(X > 7 | X > 5)$ when $n = 4$. [4]
Determine the values of $n$ for which Var$(X)$ is not defined. [5]

OCR Further Statistics 2021 June Q1

5 marks Moderate -0.3

A set of bivariate data $(X, Y)$ is summarised as follows. $n = 25$, $\Sigma x = 9.975$, $\Sigma y = 11.175$, $\Sigma x^2 = 5.725$, $\Sigma y^2 = 46.200$, $\Sigma xy = 11.575$

Calculate the value of Pearson's product-moment correlation coefficient. [1]
Calculate the equation of the regression line of $y$ on $x$. [2]

It is desired to know whether the regression line of $y$ on $x$ will provide a reliable estimate of $y$ when $x = 0.75$.

State one reason for believing that the estimate will be reliable. [1]
State what further information is needed in order to determine whether the estimate is reliable. [1]

OCR Further Statistics 2021 June Q2

4 marks Standard +0.3

The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.

Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520. [3]
Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval. [1]

OCR Further Statistics 2021 June Q3

9 marks Standard +0.3

The greatest weight $W$ N that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80. A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N.

Test at the 1% significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support. [7]
State an assumption needed in carrying out the test in part (a). [1]
Explain whether it is necessary to use the central limit theorem in carrying out the test. [1]

OCR Further Statistics 2021 June Q4

10 marks Standard +0.3

The random variable $D$ has the distribution Geo$(p)$. It is given that Var$(D) = \frac{40}{9}$. Determine

Var$(3D + 5)$. [1]
E$(3D + 5)$. [6]
$\text{P}(D > \text{E}(D))$. [3]

OCR Further Statistics 2021 June Q5

10 marks Standard +0.8

A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor $Q$ or the lectures given by Professor $R$. At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor $Q$ and a random sample of 24 students taught by Professor $R$ were ranked. The sum of the ranks of the students taught by Professor $Q$ was 726. Test at the 5% significance level whether there is a difference in the ranks of the students taught by the two professors. [10]

OCR Further Statistics 2021 June Q1

9 marks Standard +0.8

Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B. She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219.

Test at the 10\% significance level whether route A has shorter journey times than route B. [8]
State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid. [1]

OCR Further Statistics 2021 June Q2

7 marks Standard +0.8

The random variable $X$ is equally likely to take any of the $n$ integer values from $m + 1$ to $m + n$ inclusive. It is given that E$(3X) = 30$ and Var$(3X) = 36$. Determine the value of $m$ and the value of $n$. [7]

OCR Further Statistics 2021 June Q3

11 marks Challenging +1.2

26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.

Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
All 26 cards are arranged in a line, in random order.
1. Show that the probability that all the vowels are next to one another is $\frac{1}{2990}$. [3]
2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]

OCR Further Statistics 2021 June Q4

10 marks Standard +0.3

A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, $X$, up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency $f$ with which each value of $X$ is obtained. His results are shown in Table 1, together with the values of $xf$.

$x$	1	2	3	4	5	6	$\geq 7$	Total
Frequency $f$	20	15	9	13	10	10	23	100
$xf$	20	30	27	52	50	60	161	400

Table 1

State an appropriate distribution with which to model $X$, determining the value(s) of any parameter(s). [3]

Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places.

$x$	1	2	3	4	5	6	$\geq 7$
Observed frequency $O$	20	15	9	13	10	10	23
Expected frequency $E$	25	18.75	14.063	10.547	7.910	5.933	17.798
$(O-E)^2/E$	1	0.75	1.823	0.571	0.552	2.789	1.520

Table 2

Show how the expected frequency corresponding to $x \geq 7$ was obtained. [2]
Carry out the test. [5]

OCR Further Statistics 2017 Specimen Q1

6 marks Moderate -0.8

The table below shows the typical stopping distances $d$ metres for a particular car travelling at $v$ miles per hour.

$v$	20	30	40	50	60	70
$d$	13	24	36	52	72	94

State each of the following words that describe the variable $v$. Independent \quad Dependent \quad Controlled \quad Response [1]
Calculate the equation of the regression line of $d$ on $v$. [2]
Use the equation found in part (ii) to estimate the typical stopping distance when this car is travelling at 45 miles per hour. [1]

It is given that the product moment correlation coefficient for the data is 0.990 correct to three significant figures.

Explain whether your estimate found in part (iii) is reliable. [2]

Questions — OCR Further Statistics (108 questions)

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OCR Further Statistics 2018 December Q7

OCR Further Statistics 2018 December Q8

OCR Further Statistics 2021 June Q1

OCR Further Statistics 2021 June Q1

OCR Further Statistics 2021 June Q2

OCR Further Statistics 2021 June Q3

OCR Further Statistics 2021 June Q4

OCR Further Statistics 2020 November Q1

OCR Further Statistics 2020 November Q2

OCR Further Statistics 2020 November Q3

OCR Further Statistics 2020 November Q4

OCR Further Statistics 2020 November Q5

OCR Further Statistics 2020 November Q6

OCR Further Statistics 2020 November Q7

OCR Further Statistics 2020 November Q8

OCR Further Statistics 2021 June Q1

OCR Further Statistics 2021 June Q2

OCR Further Statistics 2021 June Q3

OCR Further Statistics 2021 June Q4

OCR Further Statistics 2021 June Q5

OCR Further Statistics 2021 June Q1

OCR Further Statistics 2021 June Q2

OCR Further Statistics 2021 June Q3

OCR Further Statistics 2021 June Q4

OCR Further Statistics 2017 Specimen Q1

Book	A	B	C	D	E	F	G	H	I	J	K	L
\(x\)	0.95	0.65	0.70	0.90	0.55	1.40	1.50	0.50	1.15	0.35	0.20	0.35
\(y\)	6.06	7.00	2.00	5.87	4.00	5.36	7.19	2.50	3.00	8.29	1.37	2.00

\(x\)	1	2	3	4	5	6	\(\geqslant 7\)	Total
Frequency \(f\)	20	15	9	13	10	10	23	100
\(xf\)	20	30	27	52	50	60	161	400

\(x\)	1	2	3	4	5	6	\(\geqslant 7\)
Observed frequency \(O\)	20	15	9	13	10	10	23
Expected frequency \(E\)	25	18.75	14.063	10.547	7.910	5.933	17.798
\((O - E)^2/E\)	1	0.75	1.823	0.571	0.552	2.789	1.520

\(x\)	1	2	3	4	5	6	\(\geq 7\)	Total
Frequency \(f\)	20	15	9	13	10	10	23	100
\(xf\)	20	30	27	52	50	60	161	400

\(v\)	20	30	40	50	60	70
\(d\)	13	24	36	52	72	94