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2. A factory buys 10% of its components from supplier A , 30% from supplier B and the rest from supplier C. It is known that 6% of the components it buys are faulty. Of the components bought from supplier A , 9% are faulty and of the components bought from supplier B , 3% are faulty. (a) Find the percentage of components bought from supplier C that are faulty. (3) A component is selected at random. (b) Explain why the event “the component was bought from supplier B ” is not statistically independent from the event “the component is faulty”. (1)
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2. The Venn diagram shows three events, A , B and C , and their associated probabilities.
A
y 0.
C
z x
B
Events B and C are mutually exclusive. Events A and C are independent. Showing your working, find the value of x , the value of y and the value of z. (5)
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5. Two bags, A and B , each contain balls which are either red or yellow or green.
Bag A contains 4 red, 3 yellow and n green balls. Bag B contains 5 red, 3 yellow and 1 green ball. A ball is selected at random from bag A and placed into bag B. A ball is then selected at random from bag B and placed into bag A. The probability that bag A now contains an equal number of red, yellow and green balls is p. Given that p > 0, find the possible values of n and p. (5)
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SECTION A: STATISTICS
Answer ALL questions. Write your answers in the spaces provided.
1. Helen believes that the random variable C , representing cloud cover from the large data set, can be modelled by a discrete uniform distribution. (a) Write down the probability distribution for C. (2) (b) Using this model, find the probability that cloud cover is less than 50% (1) Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than 50% was 0. (c) Comment on the suitability of Helen’s model in the light of this information. (1) (d) Suggest an appropriate refinement to Helen’s model. (1)
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5. A biased spinner can only land on one of the numbers 1, 2, 3 or 4. The random variable X represents the number that the spinner lands on after a single spin and P( X = r ) = P( X = r + 2) for r = 1, 2 Given that P( X = 2) = 0. (a) find the complete probability distribution of X. (2) Ambroh spins the spinner 60 times. (b) Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. (3)
The random variable Y = (^12) X (c) Find P( Y – X - 4) (3)
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3. A fair 5-sided spinner has sides numbered 1, 2, 3, 4 and 5
The spinner is spun once and the score of the side it lands on is recorded. (a) Write down the name of the distribution that can be used to model the score of the side it lands on. (1) The spinner is spun 28 times. The random variable X represents the number of times the spinner lands on 2 (b) (i) Find the probability that the spinner lands on 2 at least 7 times. (ii) Find P(4 - X 8) (5)
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5. Afrika works in a call centre.
She assumes that calls are independent and knows, from past experience, that on each sales call that she makes there is a probability of 1 6 that it is successful.
Afrika makes 9 sales calls. (a) Calculate the probability that at least 3 of these sales calls will be successful. (2) The probability of Afrika making a successful sales call is the same each day. Afrika makes 9 sales calls on each of 5 different days. (b) Calculate the probability that at least 3 of the sales calls will be successful on exactly 1 of these days. (2) Rowan works in the same call centre as Afrika and believes he is a more successful salesperson. To check Rowan’s belief, Afrika monitors the next 35 sales calls Rowan makes and finds that 11 of the sales calls are successful. (c) Stating your hypotheses clearly test, at the 5% level of significance, whether or not there is evidence to support Rowan’s belief. (4)
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5. Manon has two biased spinners, one red and one green.
The random variable R represents the score when the red spinner is spun. The random variable G represents the score when the green spinner is spun. The probability distributions for R and G are given below.
r 2 3 g 1 4
P( R = r ) 14 34 P( G = g ) (^2313)
Manon spins each spinner once and adds the two scores. (a) Find the probability that (i) the sum of the two scores is 7 (ii) the sum of the two scores is less than 4 (3) The random variable X = mR + nG where m and n are integers.
P( X = 20) = 16 and P( X = 50) = (^14)
(b) Find the value of m and the value of n (5)
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6. The discrete random variable X has the following probability distribution
x a b c P( X = x ) log 36 a log 36 b log 36 c
where
- a , b and c are distinct integers ( a < b < c )
- all the probabilities are greater than zero (a) Find (i) the value of a (ii) the value of b (iii) the value of c Show your working clearly. (5) The independent random variables X 1 and X 2 each have the same distribution as X
(b) Find P( X 1 = X 2 )
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3. Dian uses the large data set to investigate the Daily Total Rainfall, r mm, for Camborne.
(a) Write down how a value of 0 < r 0.05 is recorded in the large data set. (1) Dian uses the data for the 31 days of August 2015 for Camborne and calculates the following statistics
n = 31 ∑ r = 174.9 ∑ r^2 = 3523.
(b) Use these statistics to calculate (i) the mean of the Daily Total Rainfall in Camborne for August 2015, (ii) the standard deviation of the Daily Total Rainfall in Camborne for August 2015. (3) Dian believes that the mean Daily Total Rainfall in August is less in the South of the UK than in the North of the UK. The mean Daily Total Rainfall in Leuchars for August 2015 is 1.72 mm to 2 decimal places. (c) State, giving a reason, whether this provides evidence to support Dian's belief. (2) Dian uses the large data set to estimate the proportion of days with no rain in Camborne for 1987 to be 0.27 to 2 decimal places. (d) Explain why the distribution B(14, 0.27) might not be a reasonable model for the number of days without rain for a 14‑day summer event. (1)
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5. Past records show that 15% of customers at a shop buy chocolate. The shopkeeper believes that moving the chocolate closer to the till will increase the proportion of customers buying chocolate. After moving the chocolate closer to the till, a random sample of 30 customers is taken and 8 of them are found to have bought chocolate. Julie carries out a hypothesis test, at the 5% level of significance, to test the shopkeeper’s belief. Julie’s hypothesis test is shown below. H 0 : p = 0. H 1 : p. 0. Let X = the number of customers who buy chocolate. X ~ B(30, 0.15) P( X = 8) = 0. 0.0420 0.05 so reject H 0 There is sufficient evidence to suggest that the proportion of customers buying chocolate has increased. (a) Identify the first two errors that Julie has made in her hypothesis test. (2) (b) Explain whether or not these errors will affect the conclusion of her hypothesis test. Give a reason for your answer. (1) (c) Find, using a 5% level of significance, the critical region for a one-tailed test of the shopkeeper’s belief. The probability in the tail should be less than 0. (2) (d) Find the actual level of significance of this test. (1)
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4. A lake contains three different types of carp.
There are an estimated 450 mirror carp, 300 leather carp and 850 common carp. Tim wishes to investigate the health of the fish in the lake. He decides to take a sample of 160 fish. (a) Give a reason why stratified random sampling cannot be used. (1) (b) Explain how a sample of size 160 could be taken to ensure that the estimated populations of each type of carp are fairly represented. You should state the name of the sampling method used. (2) As part of the health check, Tim weighed the fish. His results are given in the table below.
Weight ( w kg) Frequency (f) Midpoint ( m kg) 2 w < 3.5 8 2. 3.5 w < 4 32 3. 4 w < 4.5 64 4. 4.5 w < 5 40 4. 5 w < 6 16 5. ( You may use (^) f m 692 and (^) f m^2 3053 ) (c) Calculate an estimate for the standard deviation of the weight of the carp. (2) Tim realised that he had transposed the figures for 2 of the weights of the fish. He had recorded in the table 2.3 instead of 3.2 and 4.6 instead of 6. (d) Without calculating a new estimate for the standard deviation, state what effect (i) using the correct figure of 3.2 instead of 2. (ii) using the correct figure of 6.4 instead of 4. would have on your estimated standard deviation. Give a reason for each of your answers. (2)
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2. A manufacturer of sweets knows that 8% of the bags of sugar delivered from supplier A will be damp. A random sample of 35 bags of sugar is taken from supplier A. (a) Using a suitable model, find the probability that the number of bags of sugar that are damp is (i) exactly 2 (ii) more than 3 (3) Supplier B claims that when it supplies bags of sugar, the proportion of bags that are damp is less than 8% The manufacturer takes a random sample of 70 bags of sugar from supplier B and finds that only 2 of the bags are damp. (b) Carry out a suitable test to assess supplier B ’s claim. You should state your hypotheses clearly and use a 10% level of significance. (4)
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4. A dentist knows from past records that 10% of customers arrive late for their appointment. A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment. A random sample of 50 of the dentist's customers is taken. (a) Write down - a null hypothesis corresponding to no change in the proportion of customers who arrive late - an alternative hypothesis corresponding to the manager's belief (1) (b) Using a 5% level of significance, find the critical region for a two‑tailed test of the null hypothesis in (a) You should state the probability of rejection in each tail, which should be less than 0. (3) (c) Find the actual level of significance of the test based on your critical region from part (b) (1) The manager observes that 15 of the 50 customers arrived late for their appointment. (d) With reference to part (b), comment on the manager’s belief. (1)
