Questions - Edexcel S2 - A-Level Maths

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Edexcel S2 Q1

5 marks Easy -2.0

A large dental practice wishes to investigate the level of satisfaction of its patients.

Suggest a suitable sampling frame for the investigation. [1]
Identify the sampling units. [1]
State one advantage and one disadvantage of using a sample survey rather than a census. [2]
Suggest a problem that might arise with the sampling frame when selecting patients. [1]

Edexcel S2 Q2

7 marks Moderate -0.8

The random variable R has the binomial distribution B(12, 0.35).

Find P(R ≥ 4). [2]

The random variable S has the Poisson distribution with mean 2.71.

Find P(S ≤ 1). [3]

The random variable T has the normal distribution N(2.5, 5²).

Find P(T ≤ 18). [2]

Edexcel S2 2004 January Q1

5 marks Easy -1.8

A large dental practice wishes to investigate the level of satisfaction of its patients.

Suggest a suitable sampling frame for the investigation. [1]
Identify the sampling units. [1]
State one advantage and one disadvantage of using a sample survey rather than a census. [2]
Suggest a problem that might arise with the sampling frame when selecting patients. [1]

Edexcel S2 2004 January Q2

7 marks Easy -1.3

The random variable $R$ has the binomial distribution B(12, 0.35).

Find P($R \geq 4$). [2]

The random variable $S$ has the Poisson distribution with mean 2.71.

Find P($S \leq 1$). [3]

The random variable $T$ has the normal distribution N(25, $5^2$).

Find P($T \leq 18$). [2]

Edexcel S2 2004 January Q3

9 marks Moderate -0.3

The discrete random variable $X$ is distributed B($n$, $p$).

Write down the value of $p$ that will give the most accurate estimate when approximating the binomial distribution by a normal distribution. [1]
Give a reason to support your value. [1]
Given that $n = 200$ and $p = 0.48$, find P($90 \leq X < 105$). [7]

Edexcel S2 2004 January Q4

10 marks Moderate -0.8

Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]

A researcher has suggested that 1 in 150 people is likely to catch a particular virus. Assuming that a person catching the virus is independent of any other person catching it,

find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [4]

Edexcel S2 2004 January Q5

13 marks Moderate -0.3

Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.

Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. [2]

Find the probability that in any randomly selected 10 minute interval

exactly 6 cars pass this point, [3]
at least 9 cars pass this point. [2]

After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.

Test, at the 5\% level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly. [6]

Edexcel S2 2004 January Q6

13 marks Standard +0.3

From past records a manufacturer of ceramic plant pots knows that 20\% of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.

Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20. The probability of rejection in either tail should be as close as possible to 2.5\%. [5]
Write down the significance level of the above test. [1]

A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.

Using a 5\% level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period. [7]

Edexcel S2 2004 January Q7

18 marks Moderate -0.3

The continuous random variable $X$ has probability density function $$\text{f}(x) = \begin{cases} kx(5 - x), & 0 \leq x \leq 4, \\ 0, & \text{otherwise,} \end{cases}$$ where $k$ is a constant.

Show that $k = \frac{3}{56}$. [3]
Find the cumulative distribution function F($x$) for all values of $x$. [4]
Evaluate E($X$). [3]
Find the modal value of $X$. [3]
Verify that the median value of $X$ lies between 2.3 and 2.5. [3]
Comment on the skewness of $X$. Justify your answer. [2]

Edexcel S2 2009 January Q1

11 marks Standard +0.3

A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3. The daisies are distributed randomly throughout the field. Find the probability that, in a randomly chosen square there will be

more than 2 daisies, [3]
either 5 or 6 daisies. [2]

The botanist decides to count the number of daisies, $x$, in each of 80 randomly selected squares within the field. The results are summarised below $$\sum x = 295 \quad \sum x^2 = 1386$$

Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places. [3]
Explain how the answers from part (c) support the choice of a Poisson distribution as a model. [1]
Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square. [2]

Edexcel S2 2009 January Q2

9 marks Easy -1.2

The continuous random variable $X$ is uniformly distributed over the interval $[-2, 7]$.

Write down fully the probability density function f(x) of $X$. [2]
Sketch the probability density function f(x) of $X$. [2]

Find

E($X^2$), [3]
P($-0.2 < X < 0.6$). [2]

Edexcel S2 2009 January Q3

7 marks Moderate -0.3

A single observation $x$ is to be taken from a Binomial distribution B(20, $p$). This observation is used to test $H_0 : p = 0.3$ against $H_1 : p \neq 0.3$

Using a 5\% level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to 2.5\%. [3]
State the actual significance level of this test. [2]

The actual value of $x$ obtained is 3.

State a conclusion that can be drawn based on this value giving a reason for your answer. [2]

Edexcel S2 2009 January Q4

12 marks Moderate -0.8

The length of a telephone call made to a company is denoted by the continuous random variable $T$. It is modelled by the probability density function $$\text{f}(t) = \begin{cases} kt & 0 \leqslant t \leqslant 10 \\ 0 & \text{otherwise} \end{cases}$$

Show that the value of $k$ is $\frac{1}{50}$. [3]
Find P($T > 6$). [2]
Calculate an exact value for E($T$) and for Var($T$). [5]
Write down the mode of the distribution of $T$. [1]

Sketch the graph of a more suitable probability density function for $T$. [1]

Edexcel S2 2009 January Q5

9 marks Moderate -0.3

A factory produces components of which 1\% are defective. The components are packed in boxes of 10. A box is selected at random.

Find the probability that the box contains exactly one defective component. [2]
Find the probability that there are at least 2 defective components in the box. [3]
Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components. [4]

Edexcel S2 2009 January Q6

14 marks Standard +0.3

A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.

1. Test, at the 10\% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
2. State the minimum number of visits required to obtain a significant result.
[7]
State an assumption that has been made about the visits to the server. [1]

In a random two minute period on a Saturday the web server is visited 20 times.

Using a suitable approximation, test at the 10\% level of significance, whether or not the rate of visits is greater on a Saturday. [6]

Edexcel S2 2009 January Q7

13 marks Standard +0.3

A random variable $X$ has probability density function given by $$\text{f}(x) = \begin{cases} -\frac{2}{9}x + \frac{8}{9} & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$

Show that the cumulative distribution function F(x) can be written in the form $ax^2 + bx + c$, for $1 \leqslant x \leqslant 4$ where $a$, $b$ and $c$ are constants. [3]
Define fully the cumulative distribution function F(x). [2]
Show that the upper quartile of $X$ is 2.5 and find the lower quartile. [6]

Given that the median of $X$ is 1.88

describe the skewness of the distribution. Give a reason for your answer. [2]

Edexcel S2 2011 January Q1

10 marks Moderate -0.3

A disease occurs in 3\% of a population.

State any assumptions that are required to model the number of people with the disease in a random sample of size $n$ as a binomial distribution. [2]
Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people. [3]
Find the mean and variance of the number of people with the disease in a random sample of 100 people. [2]

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination. [3]

Edexcel S2 2011 January Q2

6 marks Moderate -0.5

A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the 5\% level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly. [6]

Edexcel S2 2011 January Q3

11 marks Moderate -0.3

The continuous random variable $X$ is uniformly distributed over the interval $[-1,3]$. Find

E($X$) [1]
Var($X$) [2]
E($X^2$) [2]
P($X < 1.4$) [1]

A total of 40 observations of $X$ are made.

Find the probability that at least 10 of these observations are negative. [5]

Edexcel S2 2011 January Q4

6 marks Standard +0.3

Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly. [6]

Edexcel S2 2011 January Q5

13 marks Moderate -0.3

A continuous random variable $X$ has the probability density function f($x$) shown in Figure 1. \includegraphics{figure_1} Figure 1

Show that f($x$) = $4 - 8x$ for $0 \leqslant x \leqslant 0.5$ and specify f($x$) for all real values of $x$. [4]
Find the cumulative distribution function F($x$). [4]
Find the median of $X$. [3]
Write down the mode of $X$. [1]
State, with a reason, the skewness of $X$. [1]

Edexcel S2 2011 January Q6

16 marks Standard +0.3

Cars arrive at a motorway toll booth at an average rate of 150 per hour.

Suggest a suitable distribution to model the number of cars arriving at the toll booth, $X$, per minute. [2]
State clearly any assumptions you have made by suggesting this model. [2]

Using your model,

find the probability that in any given minute
1. no cars arrive,
2. more than 3 cars arrive.
[3]
In any given 4 minute period, find $m$ such that P($X > m$) = 0.0487 [3]
Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period. [6]