Questions and Answers of Cambridge International AS & A Level Further Mathematics

A charity launched a campaign to raise awareness of low pay amongst full-time workers in company canteens. The charity now believes the median weekly wage in this job sector has increased.After the
The continuous random variable X has probability density function, f, given by:The random variable Y is defined by Y = X2.a. Show that the cumulative distribution function G(y) is given by:b. Show
Trees of the same species grow on opposite sides of a river valley. The heights of eight randomly selected trees from each side of the river valley are measured and the results, in metres, are given
The owner of three driving schools, A, B and C, wished to assess whether there was an association between passing the driving test and the school attended. He selected a random sample of learner
A biased die has probability p of showing a 6. The random variable X counts the number of trials up to and including the trial in which the first 6 is obtained. The random variable I counts the
A discrete random variable, X, has the probability generating function:Gx(r) = 3/10 - 7ta. Find the probability generating function for Y = 3X + 4.b. Show that E(Y) = 11.c. Find P(Y = k) for k = 4,
<p style="text-align: justify;">A random sample of five metal rods produced by a machine is taken. Each rod is tested for hardness. The results, in suitable units, are as
Let X~Geo(0.1), X2~Geo(0.2) and X3~ Geo(0.3).Let Y = 3X1 + 2X2 + X3.a. Find Gy(t).b. Find P(Y = 7).
Let Gx(t) = - 1).a. For Y = X - 1, find Gy(1).b. For Y = 3 - 2X, find Gy(t).
A regular octagonal spinner is made up of eight isosceles triangles. Two of the triangles have a score of 2, three triangles have a score of 'a' and three triangles have a score of ‘b’, where a
A random variable X has a probability distribution, f(x), given by:a. Show that k = 3e3/e3 -1b. Find the median value of X. Ske-3* 0
Let A be a discrete random variable with probability distribution:Four independent observations are made. Y is the sum of these observations.a. Find the probability density function of Y.b. Find
Let Gx(t) = 3t/ 5 - 2ta. For Y= 5X, find Gy(t).b. For Y = X + 7, find Gy(t).
Find E(X) and Var(X) of the following probability generating function. 1+2 Gx() = (2 – P)(4 – t)
A school is considering buying a large number of safety devices to install in the classrooms. The devices are designed to activate sprinklers if the temperature in the room exceeds 65°C. A sample of
An observation from X- Geo(0.3) is taken five times: X1, ... X5.a. Find the probability generating function of Y = (X1, … X5).b. Find P(Y ≤ 8) to 4 significant figures.
Let X~ Po(a) and Y~ Po(b). Prove that if X and Y are independent:a. E(X + Y) = a + b b. Var(X + Y) = a + b
Find E(X) and Var(X) of the following probability generating function,Where q = 1 – p. Gy(i) = 1- qt
A random sample of 40 observations of a random variable X. and a random sample of 25 observations of a random variable Y. are taken. The sample means and unbiased estimates are:x̅ = 13.6 y̅ =
Consider the independent discrete random variables X, Y and Z, where:Gx(t) = (0.7 + 0.3t)3Gy(t) = (0.3 + 0.7t)4Gz(t) = 3t/10 - 7tFind the probability generating function for the following.a. A = 2X +
Let X~Bin(3, 0.2) and Y~Po(2), where X and Y are independent.a. Find the probability generating function for X + Y.b. Find the value of E(X + Y).c. Find the value of G''x+y(t). Write your answer in
In a game, the probability that player A wins on her rth go can be described as a discrete random variable X, with probability function P(X = r) = 1/2r, for r = 1, 2, 3, ….a. Find the probability
The variable Y can take only the values 1, 2, 3, ... and is such that P(Y = r) = kP(X = r), where X~ Po(λ). Show that the probability generating function of Y is given by: edt – 1 Gy(1) = et - 1
Consider the discrete random variable X, with probability distribution:and Y, with probability distribution:It is given that A = 2Y - 3X.a. Find Gx(t).b. Find Gy(t).c. Find GA(t). 3 P(X =x) 1 3 3 2.
Let X~ Geo(0.4) and Y~ Geo(0.6), where X and Y are independent.a. Write down an expression for Gx+y(t).b. Express 5/(5 – 2t)(5 – 3t) in partial fractions.c. Hence, find P(X + Y = k) for k = 2, 3,
Find E(X) and Var(X) of the following probability generating function. 9 Gr(1) = (5 – 21)?
A game consists of rolling two die, one die with six sides, the other die with eight sides, and adding the scores together. Both die are fair, with the first numbered 1 to 6 and the second numbered 1
Two independent random variables X and Y have corresponding probability generating functions:Gx(t) = e3(t – 1)andGy(t) = et(t – 1)It is given that A = Y - X.a. A student made the following
Find E(X) and Var(X) of the following probability generating function. 3+ 5t + f + f) Gx(1) =
Consider the discrete random variable X, with probability distribution:Three independent observations of X are taken and the values are added to form Y. Three independent observations of X are taken
A discrete random variable A' has probability distribution:A discrete random variable Y has probability distribution:Given that X and Y are independent, find:a. Gx(t)b. Gy(t)c. Gx+y(t) -2 -1 1 2.
For the following probability generating function:a. Find G'x(1).b. Find G''x(1).c. Find E(X).d. Find Var(X). Gx(1) =(1 + 4t + 6r2 + 41 + ) 16
The discrete random variable X is the number of times we throw a pair of fair die to get a sum of eight. Find the probability generating function, as well as the expected number of throws and the
Consider the discrete random variable X, with probability generating function:Gx(t) = t2/5(4 + t)X1 and X2 are two independent observations. Given that Y = X1 + X2, find:a. Gy(t)b. The probability
A discrete random variable X has probability distribution:A discrete random variable Y has probability distribution:Given that X and Y are independent, find:a. Gx(t) b. Gy(t)c. Gx+y(t) P(X = x)
For the following probability generating function:a. Find G'x(1).b. Find G''x(1).c. Find E(X).d. Find Var(X). Gx(t) =(1 + 21 + 31 + 24 + ) 10
A discrete random variable, X, has the probability distribution:A discrete random variable, Y, has the probability distribution:a. Find Gx(t).b. Find Gy(t)Hence, given that X and Y are independent:c.
For each of the following distributions, write down the probability generating function, Gx(t).a. X~ Bin (20, 0.3)b. X~ Bin(10, 0.25)c. X~ Bin (50, 0.04)
For each of the following distributions, write down the probability generating function, Gx(t).a. X~ Po(4) b. X~ Po (2.3) c. X~'Po(12)
For each of the following distributions, write down the probability generating function, G x(t).a. X~Geo(0.1) b. X~Geo (0.7) c. X~Geo (0.4)
Find the probability generating function for: 2 3 7. 1. P(X = x) 16 16 8. 8
Find the probability generating function for:Write your answer in the form atb(1 + t2)c. 4 6. 8 10 1 3. 1 1 P(X= x) 16 8 4 16 2.
Find the probability distribution for the following probability generating function. Gr(t) =(2 + A+ 2r") 21) 6.
Find the probability distribution for the following probability generating function. G,(1) = + 5'5,
A distribution has a probability generating function of:a. Find the probabilities for when r = 0, 1,2,3.b. Find a general formula for P(X = k), k ≥ 1. 1/2 +t) Gy() =) 3 2
Independent trials, each with the probability of 'success' p, are carried out. The random variable X counts the number of trials up to and including that on which the first success is obtained. Write
Consider the probability generating function Gx(t) = k/(5 – 2t)2. Find the value of k.
For the dataset in question 2a:a. State whether a normal approximation would be appropriate, giving a reason for your answerb. Find E(S+) and Var(S+)c. By assuming that a normal approximation would
It is believed that identical twins have similar IQ levels. Thirty pairs of identical twins participate in an IQ test and their results are recorded. There are no tied or zero ranks. The sum of
The number of accidents on a road per day is recorded for 80 days, giving the following results.It is thought that the dataset models a Poisson distribution with a rate of 2.5 accidents per day. Test
Assuming that a Wilcoxon signed-rank test is appropriate for the data, calculate T (the test statistic) based on the null hypotheses stated.a. H0: population median is 85.b. H0: population median is
For the following paired datasets, a paired sign test will be performed. Calculate S+.a.b.c. 11 10 7 13 6 9 11 12 13 7 5 G. 12
For the following pairs of data, calculate the value of T= min(P, N).a. b.c. B 6.2 6.7 7.6 7.3 5.7 5.3 6.2 6 8.4 7.8 7.2 7.3
For each of the following datasets, calculate the values of:i. Rmii. m(n + m + 1) - Rm iii. Wa.b.c.d. Y. 5.7 4.8 4.4 3.9 4.9 4.1 4.6 4.2 5.2 4.7 4.5 4.3 5.3
The dining room in a school is some distance away from the building that has all of the classrooms in it. The school believes that students take longer walking back from the dining room after lunch
For each of the following, find P(X ≤ S+|X~Bin(n, 0.5)).a. S+ = 3, n = 12b. S+ = 2, n = 9c. S+ = 4, n = 15d. S+ = 5, n = 20
For each sample size and significance level given, state the critical value for a Wilcoxon signed-rank test.a. n = 8, 5% significance, one-tailedb. n = 15, 1% significance, one-tailedc. n = 18, 2%
a. For the following dataset, state the value of n to be used in the paired sign test. Give a reason for your answer.b. Find the value of S+ for the dataset in part a. 160 158 159 159 167 158 D 166
For the hypotheses, sample size and test statistic given, state whether you would reject or not reject the null hypothesis.a. H0: the population medians are equal.H1: the population medians are not
For the corresponding dataset in question 1a-d, state the critical value for the following significance levels.a. Significance level 5% one-tailedb. Significance level 10% two-tailedc. Significance
The blood cholesterol levels of 30 males and 20 females are measured. These data are shown in the following table.Using a suitable approximation, test, at the 5% significance level, whether the blood
For each of the following, find P(X ≥ S+|X~Bin(n, 0.5)).a. S+ = 11, n = 14b. S+ = 10, n = 12c. S+ = 13, n = 15d. S+ = 17, n = 25
State the assumptions required for the use of the Wilcoxon signed-rank test.
In each case, state whether you would reject or not reject H0 at the stated significance level.H0: the population medians are equal.H1: the population medians are not equal.a. n = 9, S+ = 3, 10%
For a sample size of n = 18, a Wilcoxon signed-rank test is to be carried out. Where T = min(P, N):a. Find E(7).b. Find Var(T).c. Given that the value of T is 59 and that a normal approximation is
For the following dataset sizes, find E(Rm) and Var(Rm).a. n = 12, m = 9 b. n = 13, m = 12 c. n = 9, m = 8 d. n = 15, m = 13
An investigation is conducted into the pollution levels in a major city. The number of 2.5 mm particles can be measured using the Air Quality Index (AQI). The World Health Organization recommends
For the given hypotheses, sample size and value of S+, state whether the null hypothesis should be rejected or not rejected at the 5% significance level. Do not use a normal approximation.A. H0: The
For each of the following tests, find:• E(T) and Var(T)• The test statistic when approximating to the normal distribution. Also state whether you would reject or not reject the null hypothesis.a.
Percentage marks are obtained for a random sample of eight A Level students for their AS & A Level examinations in Statistics and in Mechanics. The following table shows their results.a. Test, at
a. For the following dataset sizes and W values, find the test statistic if a normal approximation is used. n = 18, m = 12; W = 146b. These hypotheses are given:H0: Population medians arc equal.H1:
A psychology student carries out a test on short-term memory. She shows 20 commonly used words to ten 18-year-old males. As soon as the 20 words have been shown, the psychologist asks the
A new drug to help ease bronchitis, a lung infection, is developed and needs to be inhaled using an aerosol. Two types of aerosol (X and Y) have been developed and the company wishes to test whether
The manufacturer of a brand of smartphone wishes to know what its customers think about the performance of the phone before and after introducing a new processor. The manufacturer selects, at random,
It is believed that the calorie content of a chicken sausage is different to that of a vegetarian sausage. The calorie contents of eight vegetarian sausages and seven chicken sausages are measured
Acid rain is a problem for many areas. Rain is described as being acidic if its pH level is below 5.2. In a city, a biologist takes 12 water samples and measures the pH.Test, at the 5% significance
Trials are carried out on a new tablet to help ease joint pain for people with chronic arthritis. A randomly selected sample of eight patients who have been suffering from arthritis are given the new
An eye hospital treats a large number of patients who have one eye normal, and the other eye suffers from a thinning of the cornea. Seven such patients are randomly selected and the thickness of
A particular type of tree can develop a virus that creates black spots on the leaves. The more spots, the more the virus has infected the tree. Two different virus treatments have been developed:
Plants of the same species grow on opposite sides of a river. It is thought that plants on the north side grow taller than those on the south side. Eight randomly selected plants are chosen from each
During 2017, the weekly amount of pocket money given to a random sample of 15-year-old children living in Chicago is: $9.50, $10.50, $12.50, $7.50, $20.00, $13.00, $14.00, $16.00, $17.50,
The student council of a large school believes that the average time that the A Level students spend on individual study has increased because students are more aware of the need to achieve high
A supermarket manager wishes to investigate the quality of plums. Plums can be delivered from source to the supermarket either chilled (below 5°C) or At a constant temperature of 12°C. The anion of
A website advertises used cars for sale. During September 2018, 12 cars of similar age and of the same model are for sale. The asking prices for the cars ($AUS) are:5999, 8900, 7000, 6499, 7500,
Managers at a busy international airport are studying the times taken by arriving passengers to pass immigration, collect their luggage, then pass through customs. It is known that in the past this
At a research centre, a trial is conducted to see if a new fertiliser gives a better yield of potatoes than the usual fertiliser. Ten plots of land are available for the trial. Each plot is spilt
Mr Sum wishes to investigate whether a student's test score depends on whether the test is taken in the morning or in the afternoon. He selects a random sample of 35 students of similar ability, and
The tax office claims that it takes 60 minutes to fill out their tax form. A researcher believes that it takes longer than this. A random sample of 20 people are selected, and the recorded times
For the following dataset, calculate S+ based on each of the stated null hypotheses.a. H0: population median is 150.b H0: population median is 141.c. H0: population median is 156.d. H0: population
x2 - (2 + p)x + (7 + p) = 0 has roots that differ by 1. Find the value of p given that p > 0.
a. Find E(X).b. Find E(X2).c. Find Var(X). 0
State which of the following pairs of matrices can be multiplied together, and if they can, determine the result. (i ): () () - (6 8 a b 2 6 c (5 2 1) x ( 4 0 3 8 • ( 18)- () 2. 7 5 -1 12 3 d
It is believed that some data, N = 80, can be modelled by X~Bin(6, 0.3). The table of expected values is:a. Find r.b. Find s.Write your answers to 3 decimal places. 1 3 46 E, 9.412 24.202 5.638
Find the values of:a. x28(0.9)b. x211(0.95)c. x24(0.99)d. x221(0.995)
For each of the following distributions, write the table of expected values.a.b.c. 2
a. For the following table of observed data, calculate the expected values E11, E31 and E33.b. For the following table of observed data, calculate the expected values E12, E24 and E13.Write your
A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised below:a. It is suggested that the number of letters
It is believed that some data, N = 100, can be modelled by X~Po(2.9). The table of expected values is:a. Find r.b. Find s.Write your answers to 3 decimal places. 2. 3. 4 5 6- 5.502 15.957 22.367
For the following distributions and sample sizes, write the table of expected values.a.b. Data values a, b, c, d are in the ratio 4:4:2:2; n = 240c. x = 1, 2, 3, 4 0 otherwise P(X = x) = { 30 n = 150
For each of the distributions in the question 1-ac calculate the value of the test statistic, x2.a.b.c. 2

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