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12 marks Challenging +1.8

12 A curve, $C _ { 1 }$ has equation $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 12 x + 12 } { x ^ { 2 } + 4 x - 4 }$ The line $y = k$ intersects the curve, $C _ { 1 }$ 12

1. Show that $( k + 3 ) ( k - 1 ) \geq 0$ [0pt] [5 marks]
  12
  1. (ii) Hence find the coordinates of the stationary point of $C _ { 1 }$ that is a maximum point.
    [0pt] [4 marks] 12
2. Show that the curve $C _ { 2 }$ whose equation is $y = \frac { 1 } { \mathrm { f } ( x ) }$, has no vertical asymptotes.
  [0pt] [2 marks]
  12
3. State the equation of the line that is a tangent to both $C _ { 1 }$ and $C _ { 2 }$.
  [0pt] [1 mark]

$\boldsymbol { x }$	1	2	4	9
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.2	0.4	0.35	0.05

The continuous random variable $Y$ has the following probability density function $$\mathrm { f } ( y ) = \begin{cases} \frac { 1 } { 64 } y ^ { 3 } & 0 \leq y \leq 4 \\ 0 & \text { otherwise } \end{cases}$$ Given that $X$ and $Y$ are independent, show that $\mathrm { E } \left( X ^ { 2 } + Y ^ { 2 } \right) = \frac { 1327 } { 60 }$

5 marks Moderate -0.3

4 The waiting times for patients to see a doctor in a hospital can be modelled with a normal distribution with known variance of 10 minutes. 4

A random sample of 100 patients has a total waiting time of 3540 minutes.
Calculate a $98 \%$ confidence interval for the population mean of waiting times, giving values to four significant figures.
4
Dante conducts a hypothesis test with the sample from part (a) on the waiting times. Dante's hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 38 \\ & \mathrm { H } _ { 1 } : \mu \neq 38 \end{aligned}$$ Dante uses a $2 \%$ level of significance.
Explain whether Dante accepts or rejects the null hypothesis.

5 marks Moderate -0.3

5 The diagram shows a graph of the probability density function of the random variable $X$. \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-05_574_1086_406_479} 5

State the mode of $X$.
5
Find the probability density function of $X$.

6 marks Standard +0.3

6 The discrete random variable $Y$ has the probability function $$\mathrm { P } ( Y = y ) = \begin{cases} 2 k y & y = 1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$ where $k$ is a constant. Show that $\operatorname { Var } ( 5 Y - 2 ) = 25$ \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-07_2488_1716_219_153}

8 marks Standard +0.3

7 Over a period of time it has been shown that the mean number of vehicles passing a service station on a motorway is 50 per minute. After a new motorway junction was built nearby, Xander observed that 30 vehicles passed the service station in one minute. 7

Xander claims that the construction of the new motorway junction has reduced the mean number of vehicles passing the service station per minute. Investigate Xander's claim, using a suitable test at the $1 \%$ level of significance.
7
For your test carried out in part (a) state, in context, the meaning of a Type 1 error. 7
Explain why the model used in part (a) might be invalid.

10 marks Standard +0.3

8 An insurance company groups its vehicle insurance policies into two categories, car insurance and motorbike insurance. The number of claims in a random sample of 80 policies was monitored and the results summarised in contingency Table 1. \begin{table}[h]

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

\end{table} The insurance company decides to carry out a $\chi ^ { 2 }$-test for association between number of claims and type of insurance policy using the information given in Table 1. 8

The contingency table shown in Table 2 gives some of the exact expected frequencies for this test. Complete Table 2 with the missing exact expected values. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 2}
\multirow{2}{*}{} Number of claims
0 1 2 3 or more
\multirow{2}{*}{Type of insurance policy} Car 10.0625 4.375
Motorbike 10.6875
\end{table} 8
Carry out the insurance company's test, using the $10 \%$ level of significance. \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-12_2488_1719_219_150} Additional page, if required.
Write the question numbers in the left-hand margin. Additional page, if required.
Write the question numbers in the left-hand margin. Additional page, if required.
Write the question numbers in the left-hand margin.

1 marks Easy -1.8

1 The discrete random variable $X$ has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 5 - x } { 10 } & x = 1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$ Find $\mathrm { P } ( X \geq 3 )$ Circle your answer.
0.1
0.15
0.2
0.3

1 marks Moderate -0.8

2 A binomial hypothesis test was carried out at the $5 \%$ level of significance with the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : p = 0.6 \\ & \mathrm { H } _ { 1 } : p > 0.6 \end{aligned}$$ A sample of size 30 was used to carry out the test.
Find the probability that a Type I error was made.
Circle your answer.
[0pt] [1 mark] $4.4 \%$ 4.8\% 5.0\% 9.4\%

5 marks Moderate -0.8

3 Fiona is studying the heights of corn plants on a farm. She measures the height, $x \mathrm {~cm}$, of a random sample of 200 corn plants on the farm.
The summarised results are as follows: $$\sum x = 60255 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 995$$ Calculate a $96 \%$ confidence interval for the population mean of heights of corn plants on the farm, giving your values to one decimal place.

% \(\begin{aligned			4 \text { The continuous random variable } X \text { has probability density fu }
	\qquad f ( x ) = \begin{cases} \frac { 4 } { 99 } \left( 12 x - x ^ { 2 } - x ^ { 3 } \right)	0 \leq x \leq 3
0	\text { otherwise } \end{cases} \end{aligned}\)}

8 marks Standard +0.3

$\text { Find } \mathrm { P } ( X > 1 )$ [0pt] [3 marks]
4
[0pt] [3 marks]
□
4
Find $\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)$

9 marks Standard +0.8

5 The discrete random variable $X$ has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 1 } { n } & x = 1,2 , \ldots , n \\ 0 & \text { otherwise } \end{cases}$$ 5

1. Prove that $\mathrm { E } ( X ) = \frac { n + 1 } { 2 }$ [0pt] [3 marks]
  5
  1. (ii) Prove that $\operatorname { Var } ( X ) = \frac { n ^ { 2 } - 1 } { 12 }$
    5
    State two conditions under which a discrete uniform distribution can be used to model the score when a cubic dice is rolled.
    [2 marks]

7 marks Standard +0.3

6 A company owns two machines, $A$ and $B$, which make toys. Both machines run continuously and independently. Machine $A$ makes an average of 2 errors per hour.
6

Using a Poisson model, find the probability that the machine makes exactly 5 errors in 4 hours, giving your answer to three significant figures. 6
Machine $B$ makes an average of 5 errors per hour. Both machines are switched on and run for 1 hour. The company finds the probability that the total number of errors made by machines $A$ and $B$ in 1 hour is greater than 8 . If the probability is greater than 0.4 , a new machine will be purchased.
Using a Poisson model, determine whether or not the toy company will purchase a new machine.
6
After investigation, the standard deviation of errors made by machine $A$ is found to be 0.5 errors per hour and the standard deviation of errors made by machine $B$ is also found to be 0.5 errors per hour. Explain whether or not the use of Poisson models in parts (a) and (b) is appropriate.

9 marks Standard +0.3

7 Mohammed is conducting a medical trial to study the effect of two drugs, $A$ and $B$, on the amount of time it takes to recover from a particular illness. Drug $A$ is used by one group of 60 patients and drug $B$ is used by a second group of 60 patients. The results are summarised in the table:

1 marks Easy -2.0

1 The discrete random variable $X$ has the following probability distribution

$x$	- 15	18	29
$\mathrm { P } ( X = x )$	0.2	0.7	0.1

Find $\mathrm { P } ( X > 18 )$ Circle your answer.
0.1
0.2
0.7
0.8

1 marks Moderate -0.8

2 The continuous random variable $Y$ has probability density function $\mathrm { f } ( y )$ where $$\int _ { - \infty } ^ { \infty } y \mathrm { f } ( y ) \mathrm { d } y = 16 \text { and } \int _ { - \infty } ^ { \infty } y ^ { 2 } \mathrm { f } ( y ) \mathrm { d } y = 1040$$ Find the standard deviation of $Y$ Circle your answer.
[0pt] [1 mark]
28
32
784
1024

7 marks Easy -1.2

3 The discrete random variable $A$ has the following probability distribution function $$\mathrm { P } ( A = a ) = \begin{cases} 0.45 & a = 0 \\ 0.25 & a = 1 \\ 0.3 & a = 2 \\ 0 & \text { otherwise } \end{cases}$$ 3

Find the median of $A$ 3
Find the standard deviation of $A$, giving your answer to three significant figures.
3
$\quad$ Find $\operatorname { Var } ( 9 A - 2 )$

4 marks Moderate -0.3

4 The height of lilac trees, in metres, can be modelled by a normal distribution with variance 0.7 A random sample of $n$ lilac trees is taken and used to construct a 99\% confidence interval for the population mean. This confidence interval is $( 5.239,5.429 )$ 4

Find the value of $n$ 4
Joey claims that the mean height of lilac trees is 5.3 metres.
State, with a reason, whether the confidence interval supports Joey's claim.

11 marks Standard +0.3

5 The continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} x ^ { 3 } & 0 < x \leq 1 \\ \frac { 9 } { 1696 } x ^ { 3 } \left( x ^ { 2 } + 1 \right) & 1 < x \leq 3 \\ 0 & \text { otherwise } \end{cases}$$ 5

Find $\mathrm { P } ( X < 1.8 )$, giving your answer to three decimal places.
[0pt] [3 marks]
5
Find the lower quartile of $X$
5 (d) 5
Show that $\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = \frac { 133 } { 212 }$

\multirow{2}{*}{}

Number of claims

\multirow[b]{3}{*}{Type of insurance policy}

8 marks Standard +0.3

6 The number of computers sold per day by a shop can be modelled by the random variable $Y$ where $Y \sim \operatorname { Po } ( 42 )$ 6

State the variance of $Y$ 6
One month ago, the shop started selling a new model of computer.
On a randomly chosen day in the last month, the shop sold 53 computers.
Carry out a hypothesis test, at the $5 \%$ level of significance, to investigate whether the mean number of computers sold per day has increased in the last month.
[0pt] [6 marks]
6
Describe, in the context of the hypothesis test in part (b), what is meant by a Type II error.