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CAIE S2 2016 November Q5

8 marks Standard +0.3

The masses, in grams, of certain tomatoes are normally distributed with standard deviation 9 grams. A random sample of 100 tomatoes has a sample mean of 63 grams. Find a $90 \%$ confidence interval for the population mean mass of these tomatoes.
The masses, in grams, of certain potatoes are normally distributed with known population standard deviation but unknown population mean. A random sample of potatoes is taken in order to find a confidence interval for the population mean. Using a sample of size 50 , a $95 \%$ confidence interval is found to have width 8 grams.
1. Using another sample of size 50 , an $\alpha \%$ confidence interval has width 4 grams. Find $\alpha$.
2. Find the sample size $n$, such that a $95 \%$ confidence interval has width 4 grams.

CAIE S2 2016 November Q6

9 marks Moderate -0.3

6
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_504_260_534}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_373_495_260_1123}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_497_776_534}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_367_488_778_1128} The diagrams show the probability density functions of four random variables $W , X , Y$ and $Z$. Each of the four variables takes values between 0 and 3 only, and their medians are $m _ { W } , m _ { X } , m _ { Y }$ and $m _ { Z }$ respectively.

List $m _ { W } , m _ { X } , m _ { Y }$ and $m _ { Z }$ in order of size, starting with the largest.
The probability density function of $X$ is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ (a) Show that $\mathrm { E } ( X ) = \frac { 12 } { 5 }$.
(b) Calculate $\mathrm { P } ( X > \mathrm { E } ( X ) )$.
(c) Write down the value of $\mathrm { P } ( X < 2 \mathrm { E } ( X ) )$.

CAIE S2 2016 November Q7

11 marks Standard +0.3

7 In the past the time, in minutes, taken for a particular rail journey has been found to have mean 20.5 and standard deviation 1.2. Some new railway signals are installed. In order to test whether the mean time has decreased, a random sample of 100 times for this journey are noted. The sample mean is found to be 20.3 minutes. You should assume that the standard deviation is unchanged.

Carry out a significance test, at the $4 \%$ level, of whether the population mean time has decreased. Later another significance test of the same hypotheses, using another random sample of size 100 , is carried out at the $4 \%$ level.
Given that the population mean is now 20.1, find the probability of a Type II error.
State what is meant by a Type II error in this context.

CAIE S2 2016 November Q6

9 marks Moderate -0.3

6
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_504_260_534}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_373_495_260_1123}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_497_776_534}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_367_488_778_1128} The diagrams show the probability density functions of four random variables $W , X , Y$ and $Z$. Each of the four variables takes values between 0 and 3 only, and their medians are $m _ { W } , m _ { X } , m _ { Y }$ and $m _ { Z }$ respectively.

List $m _ { W } , m _ { X } , m _ { Y }$ and $m _ { Z }$ in order of size, starting with the largest.
The probability density function of $X$ is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ (a) Show that $\mathrm { E } ( X ) = \frac { 12 } { 5 }$.
(b) Calculate $\mathrm { P } ( X > \mathrm { E } ( X ) )$.
(c) Write down the value of $\mathrm { P } ( X < 2 \mathrm { E } ( X ) )$.

CAIE S2 2016 November Q1

3 marks Easy -1.2

1 The random variable $X$ has the distribution $\operatorname { Po } ( 3.5 )$. Find $\mathrm { P } ( X < 3 )$.

CAIE S2 2016 November Q2

3 marks Easy -1.8

2 Dominic wishes to choose a random sample of five students from the 150 students in his year. He numbers the students from 1 to 150 . Then he uses his calculator to generate five random numbers between 0 and 1 . He multiplies each random number by 150 and rounds up to the next whole number to give a student number.

Dominic's first random number is 0.392 . Find the student number that is produced by this random number.
Dominic's second student number is 104 . Find a possible random number that would produce this student number.
Explain briefly why five random numbers may not be enough to produce a sample of five student numbers.

State null and alternative hypotheses for the test.
Use a binomial distribution to find the probability of a Type I error. Andre finds that 3 of the 20 packets contain free gifts.
Carry out the test.

CAIE S2 2016 November Q6

8 marks Standard +0.3

6 A variable $X$ takes values $1,2,3,4,5$, and these values are generated at random by a machine. Each value is supposed to be equally likely, but it is suspected that the machine is not working properly. A random sample of 100 values of $X$, generated by the machine, gives the following results. $$n = 100 \quad \Sigma x = 340 \quad \Sigma x ^ { 2 } = 1356$$

Find a 95\% confidence interval for the population mean of the values generated by the machine.
Use your answer to part (i) to comment on whether the machine may be working properly.

CAIE S2 2016 November Q7

9 marks Standard +0.3

7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.

Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.
Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.

CAIE S2 2016 November Q8

9 marks Standard +0.3

8
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_517_276_427}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_304_508_274_1215}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables $W , X , Y$ and $Z$. Each of the four variables takes values between - 3 and 3 only, and their standard deviations are $\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }$ and $\sigma _ { Z }$ respectively.

List $\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }$ and $\sigma _ { Z }$ in order of size, starting with the largest.
The probability density function of $X$ is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ (a) Show that $\sigma _ { X } = 2.32$ correct to 3 significant figures.
(b) Calculate $\mathrm { P } \left( X > \sigma _ { X } \right)$.
(c) Write down the value of $\mathrm { P } \left( X > 2 \sigma _ { X } \right)$.

CAIE Further Paper 4 2021 June Q1

6 marks Standard +0.3

1 Farmer $A$ grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer $B$ grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer $A$ 's trees. The masses of apples from Farmer $B$ 's trees may be assumed to be normally distributed. A random sample of 10 trees from Farmer $B$ is chosen. The masses, $x \mathrm {~kg}$, of apples produced in a year are summarised as follows. $$\sum x = 152.0 \quad \sum x ^ { 2 } = 2313.0$$ Test, at the $5 \%$ significance level, whether Farmer $B$ 's claim is justified.

CAIE Further Paper 4 2021 June Q2

7 marks Standard +0.3

2 A company is developing a new flavour of chocolate by varying the quantities of the ingredients. A random selection of 9 flavours of chocolate are judged by two tasters who each give marks out of 100 to each flavour of chocolate.

Chocolate	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$
Taster 1	72	86	75	92	98	79	87	60	62
Taster 2	84	72	74	95	85	87	82	75	68

Carry out a Wilcoxon matched-pairs signed-rank test at the $10 \%$ significance level to investigate whether, on average, there is a difference between marks awarded by the two tasters.

CAIE Further Paper 4 2021 June Q3

8 marks Standard +0.8

3 The heights, $x \mathrm {~m}$, of a random sample of 50 adult males from country $A$ were recorded. The heights, $y \mathrm {~m}$, of a random sample of 40 adult males from country $B$ were also recorded. The results are summarised as follows. $$\Sigma x = 89.0 \quad \Sigma x ^ { 2 } = 159.4 \quad \Sigma y = 67.2 \quad \Sigma y ^ { 2 } = 113.1$$ Find a 95\% confidence interval for the difference between the mean heights of adult males from country $A$ and adult males from country $B$.
$4 X$ is a discrete random variable which takes the values $0,2,4 , \ldots$. The probability generating function of $X$ is given by $$G _ { X } ( t ) = \frac { 1 } { 3 - 2 t ^ { 2 } }$$

Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
Find $\mathrm { P } ( X = 4 )$.

CAIE Further Paper 4 2021 June Q5

10 marks Standard +0.3

5 Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.

Explain how this information tends to support the manager's suspicion.
The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.
Number of breakages 0 1 2 3 4 5 6 7 or more
Frequency 21 33 46 31 23 16 10 0
The mean number of breakages calculated from this sample is 2.5.
Use the data from this larger sample to carry out a goodness of fit test, at the $10 \%$ significance level, to test the claim.

CAIE Further Paper 4 2021 June Q6

11 marks Standard +0.8

6 The continuous random variable $X$ has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 0 \leqslant x < 1 \\ \frac { 1 } { 28 } ( 8 - x ) & 1 \leqslant x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$

Find the cumulative distribution function of $X$.
Find the value of the constant $a$ such that $\mathrm { P } ( \mathrm { X } \leqslant \mathrm { a } ) = \frac { 5 } { 7 }$.
The random variable $Y$ is given by $Y = \sqrt [ 3 ] { X }$.
Find the probability density function of $Y$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 4 2022 June Q1

8 marks Standard +0.3

1 A manager is investigating the times taken by employees to complete a particular task as a result of the introduction of new technology. He claims that the mean time taken to complete the task is reduced by more than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes, before and after the introduction of the new technology are recorded in the table.

Employee	$A$	$B$	$C$	D	$E$	$F$	G	$H$	I	J
Time before new technology	10.2	9.8	12.4	11.6	10.8	11.2	14.6	10.6	12.3	11.0
Time after new technology	9.6	8.5	12.4	10.9	10.2	10.6	12.8	10.8	12.5	10.6

Test at the 10\% significance level whether the manager's claim is justified.
State an assumption that is necessary for this test to be valid.

CAIE Further Paper 4 2022 June Q2

7 marks Challenging +1.2

2 The probability generating function, $\mathrm { G } _ { Y } ( t )$, of the random variable $Y$ is given by $$G _ { Y } ( t ) = 0.04 + 0.2 t + 0.37 t ^ { 2 } + 0.3 t ^ { 3 } + 0.09 t ^ { 4 }$$

Find $\operatorname { Var } ( Y )$.
The random variable $Y$ is the sum of two independent observations of the random variable $X$.
Find the probability generating function of $X$, giving your answer as a polynomial in $t$.

CAIE Further Paper 4 2022 June Q3

8 marks Standard +0.3

3 The continuous random variable $X$ has probability density function f given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x < 2 \\ k ( 6 - x ) & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where $k$ is a constant.

Show that $k = \frac { 3 } { 40 }$.
Given that $\mathrm { E } ( X ) = 2.5$, find $\operatorname { Var } ( X )$.
Find the median value of $X$.

CAIE Further Paper 4 2022 June Q4

8 marks Standard +0.3

4 A scientist is investigating the numbers of a particular type of butterfly in a certain region. He claims that the numbers of these butterflies found per square metre can be modelled by a Poisson distribution with mean 2.5. He takes a random sample of 120 areas, each of one square metre, and counts the number of these butterflies in each of these areas. The following table shows the observed frequencies together with some of the expected frequencies using the scientist's Poisson distribution.

Number per square metre	0	1	2	3	4	5	6	$\geqslant 7$
Observed frequency	12	20	36	32	13	6	1	0
Expected frequency	9.85	24.63	30.78	25.65	$p$	8.02	3.34	$q$

Find the values of $p$ and $q$, correct to 2 decimal places.
Carry out a goodness of fit test, at the $10 \%$ significance level, to test the scientist's claim.

CAIE Further Paper 4 2022 June Q5

9 marks Standard +0.3

5 Raman is researching the heights of male giraffes in a particular region. Raman assumes that the heights of male giraffes in this region are normally distributed. He takes a random sample of 8 male giraffes from the region and measures the height, in metres, of each giraffe. These heights are as follows. $$\begin{array} { c c c c c c c c } 5.2 & 5.8 & 4.9 & 6.1 & 5.5 & 5.9 & 5.4 & 5.6 \end{array}$$

Find a $90 \%$ confidence interval for the population mean height of male giraffes in this region. [5]
Raman claims that the population mean height of male giraffes in the region is less than 5.9 metres.
Test at the $2.5 \%$ significance level whether this sample provides sufficient evidence to support Raman's claim.