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

A random sample of 200 is taken from the adult population of a town and classified by age group and preferred type of car. The results are given in the following table.Test, at the 5% significance
For the following observed data, write the table of expected data and calculate the test statistic.a.b.c.d. 10 30 20 40
For the following sets of observed (0) and expected (E) data, calculate the value of x2, the test statistic.a.b.c. 32 59 12 14 41 32 40 50 20 15 40 25
For each of the following sets of information, find: i. The number of degrees of freedom ii. The critical value. a. Number of Ei cells after combining = 9. The data are believed to fit
It is believed that the following observed data follow a binomial distribution.Let p̂ be the unbiased estimator for the proportion, p.a. Show that p̂ = 0.393 to 3 significant figures.b. Using the
For each given table of expected data, state how many degrees of freedom would be required.a. b.c. 13.44 14.84 13.72 16.64 18.37| 16.99 17.92 19.79 18.29
A random sample of 80 observations of the continuous random variable X was taken and the values are summarised in the following table.It is required to test the goodness of fit of the distribution
The following dataset shows values for what is thought to have come from the probability distribution.A test at the 5% significance level will be carried out.a. Find the expected values.b. State the
Last year, 500 students in England entered a poetry competition. Eighty of the entries were published in a book. Each student was required to state which region of England they lived in: north,
The owner of a small ski hostel records the demand for rooms during high season. a. Show that the mean demand for rooms per night is 1.72. A test is to be carried out at the 1% significance
At the end of a statistics course,110 students sit an examination. The marks are grouped into classes, as shown in the following table.It is believed that the mean for the population is 47.5.a. Show
Residents of three towns, A, B and C, are surveyed on how good their mobile phone reception is while at home, choosing from good, satisfactory or poor. A random sample of responses are gathered from
It is believed that the following data fit the modelTest this claim at the 5% significance level. - 0
For the data below, calculate the test statistic and state how many degrees of freedom are required, assuming no parameters need to be estimated.For each following question, clearly state:• Your
Let X~N(35, 42).Find:a. P(X < 30.5)b. P(30.5 ≤ X < 40.5)c. P(40.5 X ≤ 46.5)d. P(X ≥ 46.5)For each following question, clearly state:• Your hypotheses• The value of the test
Two categories X and Y are thought to be associated. The table of observed data is:a. Give the table of expected values, to 2 decimal places.b. Calculate the test statistic.c. Test, at the 5%
The population of a country is known to have blood groups 0, A, B and AB in the ratio 5:3:2: 1.220 people are randomly selected from the population of a neighbouring country. Their blood group is
150 students take a multiple-choice test consisting of six questions. The numbers of correct answers are tabulated.Test, at the 5% significance level, whether a binomial distribution is a good model
A machine is designed to cut metal into strips of length 25m, to the nearest metre. The lengths of 100 cut pieces are grouped and recorded.It is believed that the machine is equally likely to cut the
A bank manager obtains information on 150 randomly selected loans made by the bank in the previous year. The loans are classified as either good or toxic. The manager also looks at the age groups of
A company makes climbing rope, which is cut to lengths of 50m with a standard deviation of 1.5m. A sample of 150 pieces of rope is measured and the results are recorded. Test, at the 5%
A company is preparing invoices to send to their customers. Before they are sent, they are checked and the daily number of mistakes found over a two-week period are recorded.Test, at the 5% level of
For each of the following, state whether or not it is a valid probability density function, giving a reason.a.b.c.d. (x + 3) 1
Find F(x), the cumulative distribution function for: 2 (5- 3x) -4 < x < 1 f(x) = { 95 otherwise
Find the cumulative distribution function of A = X. x < 0 F(x) = 0 < x < 20. 400 1 x> 20
The time, T seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given byi. State the expected value of T.ii. Find the median value of
Sketch the following probability density functions.a.b.c. f(x) = (x + 3) -1 < x < 2 12 otherwise
Find F(x), the cumulative distribution function for: (x² - 5 8x + 18) 2 < x < 5 f(x) = otherwise
The continuous random variable X has probability density function given bya. Find E(X).b. Find E(X2).c. Find Var(X). -(7-x) 2 < x< 5 f(x) = {21 otherwise.
Find the cumulative distribution function of A = X3. x < 0 F(x) = (x' + x) 0< x < 2. 10 1 x> 2
The continuous random variable X takes values in the interval 0 ≤ x ≤ 5 only. For 0 ≤ x ≤ 5 the graph of its probability density function f consists of two straight line segments, as shown in
Find the value of k for which represents a probability density function. Skx(3 - x) 0< x
Find F(x), the cumulative distribution function for: 1 3 < x < 7 16 f(x) = %3D 1 7< x< 13 0 otherwise
The continuous random variable X has probability density function given bya. Find E(X).b. Find E(X2).c. Find Var(X).d. find SD(X) 5 < x< 8 5 f(x) = 1 8 < x< 14 15 0 otherwise.
Find the cumulative distribution function A = 3X - 22. x
The lifetime, in years, of an electrical component is the random variable T, with probability density function f given bywhere A and λ are positive constants.i. Show that A = λ.It is known that out
Find the exact value of k for whichrepresents a probability density function. Skez(x-5) 3 < x < 4 f(x) = otherwise
Find F(x), the cumulative distribution function for: 4 (x-1) 1
The continuous random variable X has probability density function given bya. Find E(X).b. Find E(X2).c. Find Var(X) 3 1< x< 6 20 f(x) = 1 (16 – x) 6
The continuous random variable X has cumulative distribution function given byFind the cumulative distribution function of:a. A = X2b. B = √X x< 1 F(x) = (x² – 1) 1< x < 2. 1 x> 2 一31
For the given probability density functionfind:a. The value of kb. P(X = 7)c. P(X < 8)d. P(X > 6) Sk(x + 1) 5
Find F(x), the cumulative distribution function for: 1 0
The continuous random variable X has probability density function given byFind E(X), giving your answer in the form a(b + Inc). 3+ f(x) = { 64 1
The continuous random variable X has cumulative distribution function given byFind the cumulative distribution function of Y = 100X2. Ox
For the given probability density functionfind:a. The value of k b. P(X < 2) c. P(1.5 ≤ X< 3.5) Sk(x2 - 2x + 3) 1
For the given probability density function:a. Find F(x)b. Calculate F(5)c. Find P(4 ≤ x < 6)d. Find in such that F(m) = 0.5. Give your answer to 2 decimal places. (12 – x) 3
The continuous random variable X has probability density function given byFind E(X(X - 1)). (23 - x) 7< x < 11 f(x) = 56 otherwise.
The continuous random variable X has probability density function given bya. Find F(x).b. Find the cumulative distribution function of Y = 3X— 2.c. Find the probability density function of Y.
Find the value of k for which represents a probability density function. kx 0
For the given probability density function:a. Find F(x), writing your answer in the form -a/10 (x – b(x – c)2, where a, b, c are positive integersb. Find P(X ≤ 4)c. Find P(X > 5). 3 (x- 2)(8
The continuous random variable X has probability density function given byFind the exact value of E(ex). 1< x < 3 f(x) = 4 otherwise.
The continuous random variable X has probability density function given by f(x)a. Find F(x).b. Find the cumulative distribution of Y = X2/4c. Find the probability density function of Y. 2 I < x< 2
Find the value of k for which represents a probability density function. (x+1) -1
For the given probability density function:a. Find F(x)b. Find P(X 6)c. Find P(X ≤ 11)d. Find a such that P(X ≥ a) = 0.75e. Find m such that F(m) = 0.5, giving your answer to 3 significant
The continuous random variable X has probability density function given byFind E(1/X2). Give your answer in the form a In b + c. 3 (x²-10x+ 22) 4 < x < 6 16 f(x) = otherwise.
The continuous random variable X has cumulative distribution function given bya. Find the cumulative distribution function of Y = 1/X.b. Find the probability density function of Y. x< 1 25 F(x) = 1
For the given probability density functionFind:a. The value of kb. P(x < 6)c. P(X > 8)d. P(6 < x ≤ 10) 4k 5
For the given probability density function:a. Find F(x)b. Find P(X > 8)c. Show that the upper quartile is 14.3, to 1 decimal place. - 18x + 83) 6 < x< 15 f(x) = {99 otherwise
The continuous random variable X has probability density function given byFind E((X - 2)2). 4 0
The continuous random variable X has probability density function given bya. Find F(x).b. Find the cumulative distribution of Y= 5 - 2X.c. Find P(Y < 2).d. Find P(-2 < Y< 2).e. Find the
For the given probability density function Find:a. The value of kb. P(X < 3)c. P(X < 4.5) (k(6x – x) 0 < x < 3 f(x) = { 9k(5 – x) 3 < x< 5, otherwise
For the given probability density function:a. Find the modeb. Show that the 40th percentile is –2.56, correct to 2 decimal places. 12 (x + 4x2 + 1) -4 < x< 1 f(x) = { 335 otherwise
The continuous random variable X has probability density function given byFind E(1/X). (5 - x) 1
A circular ink blot has radius r, described by the probability distributiona. Find F(r), the cumulative density function.b. Find G(A), the cumulative distribution function for the area of the ink
For the given cumulative distribution function:a. Show that the median is 2.32, correct to 3 significant figures.b. Find the modec. Find E(X)d. Use your answers to parts a, b and c to comment on the
For the given data, find unbiased estimates for the mean and variance.a. 12, 16, 17, 19, 13, 14, 11, 16, 19, 21, 14, 15b. 143, 154, 156, 145, 144, 132, 135, 148, 171, 124
Given the sample variance and sample size of each set of data, find the pooled estimate of variance.a. S2x = 13.2 nx = 15, S2y = 119 ny = 13b. S2x = 161.2 nx = 21, S2y = 158.7 ny
For each of the following pairs of data, find the sample mean of the difference, d̅, and the unbiased estimator of the variance of the distance, S2d. a.b.c. X 124 139 128 119 119 112 113 128
For the following confidence intervals, find the value from the t-distribution that must be used.a. 90% confidence interval, n = 6b. 90% confidence interval, n = 8c. 95% confidence interval, n = 12d.
For the following confidence intervals, find the value that must be used from the z-distribution.a. 90% confidence interval, nx = 40, ny = 60b. 95% confidence interval, nx= 30, ny = 40c. 99%
a. A gardener P claims that a new type of fruit tree produces a higher annual mass of fruit than the type that he has previously grown. The old type of tree produced 5.2 kg of fruit per tree, on
In each case, state the magnitude of the test statistic for the given value of n and stated significance level.a. n = 11, one-tailed 5%b. n = 21, one-tailed 2.5%c. n = 15, two-tailed 5%d. n = 25,
For the following pairs of data sets, find an estimate for the pooled variance.a.b. 27 19 15 19 21 18 17 16 20 28 32 31 27 26 29 30 28 14
For the given null hypotheses, find the test statistic of the following summative data.a. H0: µd = 0, µd = 0.344, s2d = 121, n = 11b. H0: µd = 0, µd = -0.688, s2d = 11.62, n = 10c. H0: µd = -5,
For the given sample sizes and values of S2x, find the value of the standard error (s/√n).a. n = 8 S2x = 9 b. n = 6 S2x=12 c. n = 9 S2x = 22d. n = 5 S2x = 4.2
For the following confidence intervals, find the value that must be used from the t-distribution.a. 90% confidence interval, nx, = 8, ny = 6b. 95% confidence interval, nx, = 14, ny = 9c. 99%
A random sample of 10 observations of a normally distributed random variable X gave the following summarised data, where x̅ denotes the sample mean.Test. at the 10% significance level, whether the
State the null and alternative hypotheses for the following tests.a. The population mean differs from 41.b. The population mean is greater than 7.3.c. The population mean has decreased from 54.2.d.
For each question, state the magnitude of the test statistic for the given values of nx and ny and stated significance level.a. nx = 8, ny = 6, one-tailed 5%b. nx = 14, ny = 10, one-tailed 2.5%c. nx
For the data in question 2a-c, state the magnitude of critical value, given the following alternate hypothesis and significance level.a. H1: µd ≠ 0, significance level 5%b. H1: µd > 0,
Given that the data comes from an underlying normal distribution, and that x̅ = 13.2, S2x = 18, find the confidence interval for the sample size stated.a. 90% confidence interval, n = 8b. 90%
By the calculating eitheras appropriate, find the stated confidence interval for µx – µy. You may assume that the data has come from an underlying normal distribution.a.90% confidence
A random sample of 50 observations of a random variable X and a random sample of 60 observations of a random variable Y are taken. The results for the sample means, x̅ and y̅, and the unbiased
For the given test statistic, sample sizes and significance levels, state whether you would reject or not reject the null hypothesis.a. Test statistic = 1.96, n = 10, 5% one-tailedb. Test statistic =
For each of the following, state the null and alternate hypotheses.a. The difference in population means is not 0.b. The population mean for X is greater than the population mean for Y.c. The
A biologist investigates the effect of a new food on Takahe male birds. Eight birds are weighed (in kg). They are then fed the new food for 14 days and weighed again.Let us assume that the weight
For the following dataset, find a 95% confidence interval.12, 15, 16, 18, 17, 15Write each end of the interval to 2 decimal places.
Two examiners are marking an examination paper, and it is believed that examiner A is more strict than examiner B. The results from several papers are added together for each examiner, and presented
A 95% confidence interval using z values is (9.642, 14.558).a. Calculate a 90% confidence interval.b. Calculate a 99% confidence interval.
In a given week, 12 babies are born in hospital. Assume that this sample came from an underlying normal population. The length of each baby is routinely measured and is listed below (in cm):49, 50,
A diet programme aims at trying to help people lose at least 2 kg in weight within five weeks of starting the programme. A sample of eight participants are asked to volunteer to take part in the
A car rental company claims that, on average, its class C-type car will use 7.1 litres of fuel per 100 km when travelling at 60 km/h. Seven class C cars are tested on a test track and their fuel use
Two newly discovered trees, X and Y, are thought to belong to the same species. Leaf measurements are made on each tree and the estimators tabulated.a. Calculate a pooled estimate for the population
A drugs manufacturer claims that the amount of paracetamol in tablets is 60mg. A sample of ten tablets is taken and the amount of paracetamol each Is recorded:59.1, 59.7, 61.0, 59.1, 60.6, 68.9,
Takahe birds are native to New Zealand and are very rare. The male birds and female birds look very similar. The only way of differentiating males from females is to measure their weights. It is
Police trainees are given a test to assess how good their memory is, After seeing ten car plates for 15 seconds each, they must write down as many as they can remember. The trainees then attend a
While on holiday, Yushan likes to stay in youth hostels. The company that owns the hostels claims that the average price of a night's stay is $43. Yushan spends one night each in six different
A psychologist wishes to investigate the effect of sleep deprivation on reaction times. Eight students volunteer to take a test, which measures their reaction time, and then re-take the test after
The weight, Xg, of a large bag of crisps is said to be normally distributed with a mean of 175g. A sample of eight bags is opened and the contents weighed. The results are listed below.173.2, 171.5,
A company that makes computers must transport them from its warehouse to the delivery centre, with one lorry delivery per day. In three weeks' time, the usual route will have roadworks stopping the

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