Questions - Edexcel S2 - A-Level Maths

he is not late at all,
he is late more than twice. In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.
Use the Normal approximation to estimate the probability that he loses his privileges. \section*{STATISTICS 2 (A)TEST PAPER 6 Page 2}

Edexcel S2 Q5

A certain type of steel is produced in a foundry. It has flaws (small bubbles) randomly distributed, and these can be detected by X-ray analysis. On average, there are 0.1 bubbles per $\mathrm { cm } ^ { 3 }$, and the number of bubbles per $\mathrm { cm } ^ { 3 }$ has a Poisson distribution.
In an ingot of $40 \mathrm {~cm} ^ { 3 }$, find
1. the probability that there are less than two bubbles,
2. the probability that there are more than 3 but less than 10 bubbles.
A new machine is being considered. Its manufacturer claims that it produces fewer bubbles per $\mathrm { cm } ^ { 3 }$. In a sample ingot of $60 \mathrm {~cm} ^ { 3 }$, there is just one bubble.
Carry out a hypothesis test at the $1 \%$ significance level to decide whether the new machine is better. State your hypotheses and conclusion carefully.

Edexcel S2 Q6

6. A random variable $X$ has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 x ^ { 2 } ( 3 - x ) } { 27 } & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

Find the mode of $X$.
Find the mean of $X$.
Specify completely the cumulative distribution function of $X$.
Deduce that the median, $m$, of $X$ satisfies the equation $m ^ { 4 } - 4 m ^ { 3 } + 13 \cdot 5 = 0$, and hence show that $1.84 < m < 1.85$.
What do these results suggest about the skewness of the distribution?

Edexcel S2 Q7

7. A corner-shop has weekly sales (in thousands of pounds), which can be modelled by the continuous random variable $X$ with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k ( x - 2 ) ( 10 - x ) & 2 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise } . \end{array}$$

Show that $k = \frac { 3 } { 256 }$ and write down the mean of $X$.
Find the standard deviation of the weekly sales.
Find the probability that the sales exceed $\pounds 8000$ in any particular week. If the sales exceed $\pounds 8000$ per week for 4 consecutive weeks, the manager gets a bonus.
Find the probability that the manager gets a bonus in February.

Edexcel S2 Q1

\begin{enumerate} \item A company that makes ropes for mountaineering wants to assess the breaking strain of its ropes.

Explain why a sample survey, and not a census, should be used.
Suggest an appropriate sampling frame. \item It is thought that a random variable $X$ has a Poisson distribution whose mean, $\lambda$, is equal to 8 . Find the critical region to test the hypothesis $\mathrm { H } _ { 0 } : \lambda = 8$ against the hypothesis $\mathrm { H } _ { 1 } : \lambda < 8$, working at the $1 \%$ significance level. \item A child cuts a 30 cm piece of string into two parts, cutting at a random point.

Edexcel S2 Q6

When a park is redeveloped, it is claimed that $70 \%$ of the local population approve of the new design. Assuming this to be true, find the probability that, in a group of 10 residents selected at random,
1. 6 or more approve,
2. exactly 7 approve.
A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.
Use this information to carry out a hypothesis test on the original claim, working at the $5 \%$ significance level. State your conclusion clearly. If the conservationists are right, and only $45 \%$ approve of the new park,
use a suitable approximation to the binomial distribution to estimate the probability that in a larger survey, of 500 people, less than half will approve.

Edexcel S2 Q7

7. A continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x } { 3 } & 0 \leq x < 1
\mathrm { f } ( x ) = 1 - \frac { x } { 3 } & 1 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

Sketch the graph of $\mathrm { f } ( x )$ for all $x$.
Find the mean of $X$.
Find the standard deviation of $X$.
Show that the cumulative distribution function of $X$ is given by $$\mathrm { F } ( x ) = \frac { x ^ { 2 } } { 3 } \quad 0 \leq x < 1$$ and find $\mathrm { F } ( x )$ for $1 \leq x \leq 3$.

Edexcel S2 Q1

(a) Briefly explain the difference between a one-tailed test and a two-tailed test.
(b) State, with a reason, which type of test would be more appropriate to test the claim that this decade's average temperature is greater than the last decade's.
(a) Give one advantage and one disadvantage of
1. a sample survey,
2. a census.
  (b) Suggest a situation in which each could be used.
3. A pharmaceutical company produces an ointment for earache that, in $80 \%$ of cases, relieves pain within 6 hours. A new drug is tried out on a sample of 25 people with earache, and 24 of them get better within 6 hours.
  (a) Test, at the $5 \%$ significance level, the claim that the new treatment is better than the old one. State your hypotheses carefully.
A rival company suggests that the sample does not give a conclusive result;
(b) Might they be right, and how could a more conclusive statement be achieved?

Edexcel S2 Q4

4. A centre for receiving calls for the emergency services gets an average of $3 \cdot 5$ emergency calls every minute. Assuming that the number of calls per minute follows a Poisson distribution,

find the probability that more than 6 calls arrive in any particular minute. Each operator takes a mean time of 2 minutes to deal with each call, and therefore seven operators are necessary to cope with the average demand.
Find how many operators are required for there to be a $99 \%$ probability that a call can be dealt with immediately. It is found from experience that a major disaster creates a surge of emergency calls. Taking the null hypothesis $\mathrm { H } _ { 0 }$ that there is no disaster,
find the number of calls that need to be received in one minute to disprove $\mathrm { H } _ { 0 }$ at the $0.1 \%$ significance level.

Edexcel S2 Q5

5. The random variable $X$ has a continuous uniform distribution on the interval $a \leq X \leq 3 a$.

Without assuming any standard results, prove that $\mu$, the mean value of $X$, is equal to $2 a$ and derive an expression for $\sigma ^ { 2 }$, the variance of $X$, in terms of $a$.
Find the probability that $| X - \mu | < \sigma$ and compare this with the same probability when $x$ is modelled by a Normal distribution with the same mean and variance. \section*{STATISTICS 2 (A) TEST PAPER 8 Page 2}

Edexcel S2 Q6

Two people are playing darts. Peg hits points randomly on the circular board, whose radius is $a$. If the distance from the centre $O$ of the point that she hits is modelled by the variable $R$,
1. explain why the cumulative distribution function $\mathrm { F } ( r )$ is given by
$$\begin{array} { l l } \mathrm { F } ( r ) = 0 & r < 0 ,
\mathrm {~F} ( r ) = \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leq r \leq a ,
\mathrm {~F} ( r ) = 1 & r > a . \end{array}$$
By first finding the probability density function of $R$, show that the mean distance from $O$ of the points that Peg hits is $\frac { 2 a } { 3 }$. Bob, a more experienced player, aims for $O$, and his points have a distance $X$ from $O$ whose cumulative distribution function is $$\mathrm { F } ( x ) = 0 , x < 0 ; \quad \mathrm { F } ( x ) = \frac { x } { a } \left( 2 - \frac { x } { a } \right) , 0 \leq x \leq a ; \quad \mathrm { F } ( x ) = 1 , x > a .$$
Find the probability density function of $X$, and explain why it shows that Bob is aiming for $O$.

Edexcel S2 Q7

7. In an orchard, all the trees are either apple or pear trees. There are four times as many apple trees as pear trees. Find the probability that, in a random sample of 10 trees, there are

equal numbers of apple and pear trees,
more than 7 apple trees. In a sample of 60 trees in the orchard,
find the expected number of pear trees.
Calculate the standard deviation of the number of pear trees and compare this result with the standard deviation of the number of apple trees.
Find the probability that exactly 35 in the sample of 60 trees are pear trees.
Find an approximate value for the probability that more than 15 of the 60 trees are pear trees.

Edexcel S2 Q1

(a) Explain the difference between a discrete and a continuous variable.

A random number generator on a calculator generates numbers, $X$, to 3 decimal places, in the range 0 to 1 , e.g. 0.386 . The variable $X$ may be modelled by a continuous uniform distribution, having the probability density function $\mathrm { f } ( x )$, where $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$ (b) Explain why this model is not totally accurate.
(c) Sketch the cumulative distribution function of $X$.

Edexcel S2 Q2

2. A video rental shop needs to find out whether or not videos have been rewound when they are returned; it will do this by taking a sample of returned videos

State one advantage and one disadvantage of taking a sample.
Suggest a suitable sampling frame.
Describe the sampling units.
Criticise the sampling method of looking at just one particular shelf of videos.

Edexcel S2 Q3

3. The random variable $X$ is modelled by a binomial distribution $\mathrm { B } ( n , p )$, with $n = 20$ and $p$ unknown. It is suspected that $p = 0 \cdot 4$.

Find the critical region for the test of $\mathrm { H } _ { 0 } : p = 0.4$ against $\mathrm { H } _ { 1 } : p \neq 0.4$, at the $5 \%$ significance level.
Find the critical region if, instead, the alternative hypothesis is $\mathrm { H } _ { 1 } : p < 0.4$.

Edexcel S2 Q4

4. A random variable $X$ has the distribution $\mathrm { B } ( 80,0.375 )$.

Write down the mean and variance of $X$.
Use the Normal approximation to the binomial distribution to estimate $\mathrm { P } ( X > 40 )$.

Edexcel S2 Q5

5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results:

Number of lorries in

five-minute interval, $X$

Number of intervals

Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2}

continued ...
1. Show that the mean of $X$ is 3 , and find the variance of $X$.
2. Give two reasons for thinking that $X$ can be modelled by a Poisson distribution. (2 marks)
After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
Test whether the group's claim is valid. Work at the $5 \%$ significance level, and state your hypotheses clearly.

Edexcel S2 Q6

6. In a particular parliamentary constituency, the percentage of Conservative voters at the last election was $35 \%$, and the percentage who voted for the Monster Raving Loony party was $2 \%$.

Find the probability that a random sample of 10 electors includes at least two Conservative voters. Use suitable approximations to find
the probability that a random sample of 500 electors will include at least 200 who voted either Conservative or Monster Raving Loony,
the probability that a random sample of 200 electors will have at least 5 Monster Raving Loony voters in it.
One of (b) or (c) requires an adjustment to be made before a calculation is done. Explain what this adjustment is, and why it is necessary.

Edexcel S2 Q7

7. The fraction of sky covered by cloud is modelled by the random variable $X$ with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 0
\mathrm { f } ( x ) = k x ^ { 2 } ( 1 - x ) & 0 \leq x \leq 1 ,
\mathrm { f } ( x ) = 0 & x > 1 . \end{array}$$

Find $k$ and sketch the graph of $\mathrm { f } ( x )$.
Find the mean and the variance of $X$.
Find the cumulative distribution function $\mathrm { F } ( x )$.
Given that flying is prohibited when $85 \%$ of the sky is covered by cloud, show that cloud conditions allow flying nearly $90 \%$ of the time.

Edexcel S2 Q1

Briefly explain what is meant by
1. a statistical model,
  (2 marks)
2. a sampling frame,
3. a sampling unit.
4. (a) Explain what is meant by the critical region of a statistical test.
5. Under a hypothesis $\mathrm { H } _ { 0 }$, an event $A$ can happen with probability $4 \cdot 2 \%$. The event $A$ does then happen. State, with justification, whether $\mathrm { H } _ { 0 }$ should be accepted or rejected at the $5 \%$ significance level.

Edexcel S2 Q3

Briefly describe the main features of a binomial distribution. I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52 .
Explain why the distribution of $X$, the number of hearts obtained, is not $\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)$.
(2 marks)
After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is $\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)$, find
the probability of getting no hearts,
the probability of getting 4 or more hearts.
If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn.

Edexcel S2 Q4

4. A Geiger counter is observed in the presence of a radioactive source. In 100 one-minute intervals, the number of counts recorded are as follows:

No of counts, $X$	0	1	2	3	4	5	6
Frequency	10	24	29	16	12	6	3

Find the mean and variance of this data, and show that it supports the idea that the random variable $X$ is following a Poisson distribution.
Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute.
Find the number of one-minute intervals, in the sample of 100 , in which more than 6 counts would be expected. \section*{STATISTICS 2 (A) TEST PAPER 10 Page 2}

Edexcel S2 Q5

A continuous random variable $X$ has the cumulative distribution function

$$\begin{array} { l l } \mathrm { F } ( x ) = 0 & x < 2 ,
\mathrm {~F} ( x ) = k ( x - a ) ^ { 2 } & 2 \leq x \leq 6 ,
\mathrm {~F} ( x ) = 1 & x \geq 6 . \end{array}$$

Find the values of the constants $a$ and $k$.
Show that the median of $X$ is $2 ( 1 + \sqrt { 2 } )$.
Given that $X > 4$, find the probability that $X > 5$.

Edexcel S2 Q6

6. A small opinion poll shows that the Trendies have a $10 \%$ lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a $10 \%$ lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.

At the $5 \%$ significance level, test which is right, stating your null hypothesis carefully.
If it is indeed true that the Trendies are supported by $55 \%$ of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes.

Edexcel S2 Q7

7. A continuous random variable $X$ has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 6 x } { 175 } & 0 \leq x < 5
\mathrm { f } ( x ) = \frac { 6 x ( 10 - x ) } { 875 } & 5 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$

Verify that f is a probability density function.
Write down the probability that $X < 1$.
Find the cumulative distribution function of $X$, carefully showing how it changes for different domains.
Find the probability that $2 < X < 7$.

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