Edexcel S2 (Statistics 2)

The small village of Tornep has a preservation society which is campaigning for a new by-pass to be built. The society needs to measure

1. the strength of opinion amongst the residents of Tornep for the scheme and
2. the flow of traffic through the village on weekdays. The society wants to know whether to use a census or a sample survey for each of these measures.
   1. In each case suggest which they should use and specify a suitable sampling frame. [4] For the measurement of traffic flow through Tornep,
   2. suggest a suitable statistic and a possible statistical model for this statistic. [2]

Explain what you understand by

1. a population, [1]
2. a statistic. [2]

A questionnaire concerning attitudes to classes in a college was completed by a random sample of 50 students. The students gave the college a mean approval rating of 75\%.

1. Identify the population and the statistic in this situation. [2]
2. Explain what you understand by the sampling distribution of this statistic. [2]

The manager of a leisure club is considering a change to the club rules. The club has a large membership and the manager wants to take the views of the members into consideration before deciding whether or not to make the change.

1. Explain briefly why the manager might prefer to use a sample survey rather than a census to obtain the views. [2]
2. Suggest a suitable sampling frame. [1]
3. Identify the sampling units. [1]

An engineer measures, to the nearest cm, the lengths of metal rods.

1. Suggest a suitable model to represent the difference between the true lengths and the measured lengths. [2]
2. Find the probability that for a randomly chosen rod the measured length will be within 0.2 cm of the true length. [2]

Two rods are chosen at random.

1. Find the probability that for both rods the measured lengths will be within 0.2 cm of their true lengths. [2]

Explain briefly what you understand by

1. a statistic, [2]
2. a sampling distribution. [2]

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

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

On a stretch of motorway accidents occur at a rate of 0.9 per month.

1. Show that the probability of no accidents in the next month is 0.407, to 3 significant figures. [1] Find the probability of
2. exactly 2 accidents in the next 6 month period, [3]
3. no accidents in exactly 2 of the next 4 months. [3]

The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the new salesman has increased house sales. [7]

A random sample \(X_1, X_2, ..., X_n\) is taken from a finite population. A statistic Y is based on this sample.

1. Explain what you understand by the statistic Y. [2]
2. Give an example of a statistic. [1]
3. Explain what you understand by the sampling distribution of Y. [2]

A single observation x is to be taken from a Poisson distribution with parameter \(\lambda\). This observation is to be used to test H₀: \(\lambda\) = 7 against H₁: \(\lambda\) ≠ 7.

1. Using a 5\% significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to 2.5\%. [5]
2. Write down the significance level of this test. [1]

The actual value of x obtained was 5.

1. State a conclusion that can be drawn based on this value. [2]

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable Y ~ Po(30).

1. Estimate P(Y > 28). [6]

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

1. Find P(R ≥ 4). [2]

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

1. Find P(S ≤ 1). [3]

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

1. Find P(T ≤ 18). [2]

In a sack containing a large number of beads \(\frac{1}{4}\) are coloured gold and the remainder are of different colours. A group of children use some of the beads in a craft lesson and do not replace them. Afterwards the teacher wishes to know whether or not the proportion of gold beads left in the sack has changed. She selects a random sample of 20 beads and finds that 2 of them are coloured gold. Stating your hypotheses clearly test, at the 10\% level of significance, whether or not there is evidence that the proportion of gold beads has changed. [7]

An airline knows that overall 3\% of passengers do not turn up for flights. The airline decides to adopt a policy of selling more tickets than there are seats on a flight. For an aircraft with 196 seats, the airline sold 200 tickets for a particular flight.

1. Write down a suitable model for the number of passengers who do not turn up for this flight after buying a ticket. [2]

By using a suitable approximation, find the probability that

1. more than 196 passengers turn up for this flight, [3]
2. there is at least one empty seat on this flight. [2]

The continuous random variable R is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that E(R) = 3 and Var(R) = \(\frac{4}{3}\), find

1. the value of \(\alpha\) and the value of \(\beta\), [7]
2. P(R < 6.6). [2]

A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.

1. Write down two conditions that must apply for this model to be applicable. [2]

Assuming this model and a mean of 0.7 weeds per m², find

1. the probability that in a randomly chosen plot of size 4 m² there will be fewer than 3 of these weeds, [4]
2. Using a suitable approximation, find the probability that in a plot of 100 m² there will be more than 66 of these weeds. [6]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable X, the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of X. [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(X) and Var (X). [3]

A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that 20\% of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

1. at least 3, [2]
2. fewer than 2. [2]

One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,

1. use a suitable approximation to find the probability that there are enough first class stamps, [7]
2. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid. [1]

Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable X represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of X and specify it fully. [2]
2. Calculate the mean time of arrival of the bus. [3]
3. Find the cumulative distribution function of X. [4]

Jean will be late for work if the bus arrives after 8.05 a.m.

1. Find the probability that Jean is late for work. [2]

Past records show that 20\% of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps were taken and 2 of them had bought them in single packets.

1. Use these data to test, at the 5\% level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly. [6]

At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03. To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.

1. Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. Write down the significance level of this test. [1]

The continuous random variable X has cumulative distribution function $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{4}x²(4 - x²), & 0 \leq x \leq 1, \\ 1, & x > 1. \end{cases}$$

1. Find P(X > 0.7). [2]
2. Find the probability density function f(x) of X. [2]
3. Calculate E(X) and show that, to 3 decimal places, Var(X) = 0.057. [6]

One measure of skewness is $$\frac{\text{Mean} - \text{Mode}}{\text{Standard deviation}}$$

1. Evaluate the skewness of the distribution of X. [4]

1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]

A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that

1. the first 5 will occur on the sixth throw, [8]
2. in the first eight throws there will be exactly three 5s.

The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week. Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is

1. exactly 4, [2]
2. more than 5. [2]

Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.

1. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. [3]

The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.

1. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased. [5]

An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.

1. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour. [2]

Find the probability that in a randomly chosen hour

1. all Internet users connect at their first attempt, [2]
2. more than 4 users fail to connect at their first attempt. [2]

1. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period. [1]
2. Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period. [6]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length L where L ~ N(\(\mu\), 0.3²).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05. Eggs are packed in boxes of 12. Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one, [3]
2. more than three. [2]

A customer bought three boxes.

1. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. [3]

The farmer delivered 10 boxes to a local shop.

1. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. [4]

The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g.

1. Find the probability that a randomly chosen egg weighs more than 68 g. [3]

A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml. The random variable X is the volume of lemonade dispensed into a cup.

1. Specify the probability density function of X and sketch its graph. [4]

Find the probability that the machine dispenses

1. less than 183 ml, [3]
2. exactly 183 ml. [1]
3. Calculate the inter-quartile range of X. [3]
4. Determine the value of s such that P(X ≤ s) = 1 - 2P(X ≤ s). [2]
5. Interpret in words your value of s.

The continuous random variable X has cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0, & x < 1 \\ \frac{1}{2}(-x^3 + 6x^2 - 5), & 1 \leq x \leq 4 \\ 1, & x > 4 \end{cases}$$

1. Find the probability density function f(x). [3]
2. Find the mode of X. [2]
3. Sketch f(x) for all values of x. [3]
4. Find the mean \(\mu\) of X. [3]
5. Show that F(\(\mu\)) > 0.5. [1]
6. Show that the median of X lies between the mode and the mean. [2]

The owner of a small restaurant decides to change the menu. A trade magazine claims that 40\% of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.

1. Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners X who choose organic foods. [2]
2. Find P(5 < X < 15). [4]
3. Find the mean and standard deviation of X. [3]

The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly. [5]

From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m.

1. Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. [2]

The twine is usually sold in balls of length 100 m. A customer buys three balls of twine.

1. Find the probability that only one of them will have fewer than 6 faults. [6]

As a special order a ball of twine containing 500 m is produced.

1. Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive. [6]

A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that 40\% of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine. [1]
2. Suggest a suitable sampling frame for the survey. [1]
3. Identify the sampling units. [1]
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. [2]

As a pilot study the editor took a random sample of 25 subscribers.

1. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. [3]

In fact only 6 subscribers agreed to the name being changed.

1. Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the percentage agreeing to the change is less that the editor believes. [5]

The full survey is to be carried out using 200 randomly chosen subscribers.

1. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. [7]

A doctor expects to see, on average, 1 patient per week with a particular disease.

1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer. [3]
2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. [4]

The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. [6]

Medical research into the nature of the disease discovers that it can be passed from one patient to another.

1. Explain whether or not this research supports your choice of model. Give a reason for your answer. [2]

In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key 7 s after the star first appears, a simple model of the game assumes that T is a continuous uniform random variable defined over the interval [0, 1].

1. Write down P(T < 0.2). [1]
2. Write down E(T). [1]
3. Use integration to find Var(T). [4]

A group of 20 children each play this game once.

1. Find the probability that no more than 4 children stop the star in less than 0.2 s. [3]

The children are allowed to practise this game so that this continuous uniform model is no longer applicable.

1. Explain how you would expect the mean and variance of T to change. [2]

It is found that a more appropriate model of the game when played by experienced players assumes that T has a probability density function g(t) given by $$g(t) = \begin{cases} 4t, & 0 \leq t \leq 0.5, \\ 4 - 4t, & 0.5 \leq t \leq 1, \\ 0, & otherwise. \end{cases}$$

1. Using this model show that P(T < 0.2) = 0.08. [2]

A group of 75 experienced players each played this game once.

1. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s. [4]

A continuous random variable X has cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ kx^2 + 2kx, & 0 \leq x \leq 2, \\ 8k, & x > 2. \end{cases}$$

1. Show that \(k = \frac{1}{8}\). [1]
2. Find the median of X. [3]
3. Find the probability density function f(x). [3]
4. Sketch f(x) for all values of x. [3]
5. Write down the mode of X. [1]
6. Find E(X). [3]
7. Comment on the skewness of this distribution. [2]

The continuous random variable X has probability density function $$f(x) = \begin{cases} \frac{x}{15}, & 0 \leq x \leq 2, \\ \frac{x}{15}, & \\ \frac{2x}{45}, & 2 < x < 7, \\ \frac{2}{9}, & 7 \leq x \leq 10, \\ 0, & otherwise. \end{cases}$$

1. Sketch f(x) for all values of x. [3]
2. Find expressions for the cumulative distribution function, F(x), for 0 ≤ x ≤ 2 and for 7 ≤ x ≤ 10. [8]
3. Find P(X ≤ 8.2). [2]
4. Find, to 3 significant figures, E(X). [4]

A continuous random variable X has probability density function f(x) where $$f(x) = \begin{cases} k(x^3 + 2x + 1), & -1 \leq x \leq 0, \\ 0, & otherwise \end{cases}$$ where k is a positive integer.

1. Show that k = 3. [4]

Find

1. E(X), [4]
2. the cumulative distribution function F(x), [4]
3. P(−0.3 < X < 0.3). [3]