Questions — Edexcel S4 (191 questions)

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Edexcel S4 2002 June Q3
10 marks Standard +0.8
A technician is trying to estimate the area \(\mu^2\) of a metal square. The independent random variables \(X_1\) and \(X_2\) are each distributed \(\text{N}(\mu, \sigma^2)\) and represent two measurements of the sides of the square. Two estimators of the area, \(A_1\) and \(A_2\), are proposed where $$A_1 = X_1X_2 \text{ and } A_2 = \left(\frac{X_1 + X_2}{2}\right)^2.$$ [You may assume that if \(X_1\) and \(X_2\) are independent random variables then $$\text{E}(X_1X_2) = \text{E}(X_1)\text{E}(X_2)$$]
  1. Find \(\text{E}(A_1)\) and show that \(\text{E}(A_2) = \mu^2 + \frac{\sigma^2}{2}\). [4]
  2. Find the bias of each of these estimators. [2]
The technician is told that \(\text{Var}(A_1) = \sigma^4 + 2\mu^2\sigma^2\) and \(\text{Var}(A_2) = \frac{1}{2}\sigma^4 + 2\mu^2\sigma^2\). The technician decided to use \(A_1\) as the estimator for \(\mu^2\).
  1. Suggest a possible reason for this decision. [1]
A statistician suggests taking a random sample of \(n\) measurements of sides of the square and finding the mean \(\overline{X}\). He knows that \(\text{E}(\overline{X}^2) = \mu^2 + \frac{\sigma^2}{n}\) and $$\text{Var}(\overline{X}^2) = \frac{2\sigma^4}{n^2} + \frac{4\sigma^2\mu^2}{n}.$$
  1. Explain whether or not \(\overline{X}^2\) is a consistent estimator of \(\mu^2\). [3]
Edexcel S4 2002 June Q4
12 marks Standard +0.3
A recent census in the U.K. revealed that the heights of females in the U.K. have a mean of 160.9 cm. A doctor is studying the heights of female Indians in a remote region of South America. The doctor measured the height, \(x\) cm, of each of a random sample of 30 female Indians and obtained the following statistics. $$\Sigma x = 4400.7, \quad \Sigma x^2 = 646904.41.$$ The heights of female Indians may be assumed to follow a normal distribution. The doctor presented the results of the study in a medical journal and wrote 'the female Indians in this region are more than 10 cm shorter than females in the U.K.'
  1. Stating your hypotheses clearly and using a 5% level of significance, test the doctor's statement. [6]
The census also revealed that the standard deviation of the heights of U.K. females was 6.0 cm.
  1. Stating your hypotheses clearly test, at the 5% level of significance, whether or not there is evidence that the variance of the heights of female Indians is different from that of females in the U.K. [6]
Edexcel S4 2002 June Q5
13 marks Standard +0.3
The times, \(x\) seconds, taken by the competitors in the 100 m freestyle events at a school swimming gala are recorded. The following statistics are obtained from the data.
No. of competitorsSample Mean \(\overline{x}\)\(\Sigma x^2\)
Girls883.1055746
Boys788.9056130
Following the gala a proud parent claims that girls are faster swimmers than boys. Assuming that the times taken by the competitors are two independent random samples from normal distributions,
  1. test, at the 10% level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly. [7]
  2. Stating your hypotheses clearly, test the parent's claim. Use a 5% level of significance. [6]
Edexcel S4 2002 June Q6
13 marks Standard +0.8
A nutritionist studied the levels of cholesterol, \(X\) mg/cm³, of male students at a large college. She assumed that \(X\) was distributed \(\text{N}(\mu, \sigma^2)\) and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of \(\mu\) and \(\sigma^2\) as $$\hat{\mu} = 1.68, \quad \hat{\sigma}^2 = 1.79.$$
  1. Find a 95% confidence interval for \(\mu\). [4]
  2. Obtain a 95% confidence interval for \(\sigma^2\). [5]
A cholesterol reading of more than 2.5 mg/cm³ is regarded as high.
  1. Use appropriate confidence limits from parts \((a)\) and \((b)\) to find the lowest estimate of the proportion of male students in the college with high cholesterol. [4]
Edexcel S4 2002 June Q7
16 marks Standard +0.3
A proportion \(p\) of the items produced by a factory is defective. A quality assurance manager selects a random sample of 5 items from each batch produced to check whether or not there is evidence that \(p\) is greater than 0.10. The criterion that the manager uses for rejecting the hypothesis that \(p\) is 0.10 is that there are more than 2 defective items in the sample.
  1. Find the size of the test. [2]
Table 1 gives some values, to 2 decimal places, of the power function of this test.
\(p\)0.150.200.250.300.350.40
Power0.03\(r\)0.100.160.240.32
  1. Find the value of \(r\). [3]
One day the manager is away and an assistant checks the production by random sample of 10 items from each batch produced. The hypothesis that \(p = 0.10\) is rejected if more than 4 defectives are found in the sample.
  1. Find P(Type I error) using the assistant's test. [2]
Table 2 gives some values, to 2 decimal places, of the power function for this test.
\(p\)0.150.200.250.300.350.40
Power0.010.030.080.150.25\(s\)
  1. Find the value of \(s\). [1]
  2. Using the same axes, draw the graphs of the power functions of these two tests. [4]
    1. State the value of \(p\) where these graphs cross.
    2. Explain the significance if \(p\) is greater than this value.
    [2]
The manager studies the graphs in part \((e)\) but decides to carry on using the test based on a sample of size 5.
  1. Suggest 2 reasons why the manager might have made this decision. [2]
Edexcel S4 2003 June Q1
6 marks Standard +0.3
A beach is divided into two areas \(A\) and \(B\). A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area \(A\) and the unbiased estimate for the population variance is \(s_A^2 = 0.495\) mm\(^2\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s_B^2 = 1.04\) mm\(^2\).
  1. Stating your hypotheses clearly test, at the 10\% significance level, whether or not there is a difference in variability of pebble length between area \(A\) and area \(B\). [5]
  2. State the assumption you have made about the populations of pebble lengths in order to carry out the test. [1]
Edexcel S4 2003 June Q3
9 marks Challenging +1.2
A train company claims that the probability \(p\) of one of its trains arriving late is 10\%. A regular traveller on the company's trains believes that the probability is greater than 10\% and decides to test this by randomly selecting 12 trains and recording the number \(X\) of trains that were late. The traveller sets up the hypotheses H\(_0\): \(p = 0.1\) and H\(_1\): \(p > 0.1\) and accepts the null hypothesis if \(x \leq 2\).
  1. Find the size of the test. [1]
  2. Show that the power function of the test is $$1 - (1 - p)^{10}(1 + 10p + 55p^2).$$ [4]
  3. Calculate the power of the test when
    1. \(p = 0.2\),
    2. \(p = 0.6\). [3]
  4. Comment on your results from part (c). [1]
Edexcel S4 2003 June Q5
11 marks Standard +0.3
  1. Define
    1. a Type I error,
    2. a Type II error. [2]
A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.
  1. Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
  2. Calculate the probability of the Type I error for this test. [3]
  3. Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]
Edexcel S4 2003 June Q6
14 marks Standard +0.3
A random sample of three independent variables \(X_1\), \(X_2\) and \(X_3\) is taken from a distribution with mean \(\mu\) and variance \(\sigma^2\).
  1. Show that \(\frac{2}{5}X_1 - \frac{1}{5}X_2 + \frac{4}{5}X_3\) is an unbiased estimator for \(\mu\). [3]
An unbiased estimator for \(\mu\) is given by \(\hat{\mu} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.
  1. Show that Var(\(\hat{\mu}\)) = \((2a^2 - 2a + 1)\sigma^2\). [6]
  2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{\mu}\) has minimum variance. [5]
Edexcel S4 2003 June Q7
17 marks Standard +0.3
Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml, are given in the table.
Orange12345678
Method A2930262526222328
Method B2725282423262225
One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.
  1. Stating your hypotheses clearly and using a 5\% significance level, carry out this test. (You may assume \(\bar{x}_A = 26.125\), \(s_A^2 = 7.84\), \(\bar{x}_B = 25\), \(s_B^2 = 4\) and \(\sigma_A^2 = \sigma_B^2\)) [7]
Another statistician suggests analysing these data using a paired \(t\)-test.
  1. Using a 5\% significance level, carry out this test. [9]
  2. State which of these two tests you consider to be more appropriate. Give a reason for your choice. [1]
Edexcel S4 2012 June Q1
9 marks Standard +0.3
A medical student is investigating whether there is a difference in a person's blood pressure when sitting down and after standing up. She takes a random sample of 12 people and measures their blood pressure, in mmHg, when sitting down and after standing up. The results are shown below.
Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Sitting down135146138146141158136135146161119151
Standing up131147132140138160127136142154130144
The student decides to carry out a paired \(t\)-test to investigate whether, on average, the blood pressure of a person when sitting down is more than their blood pressure after standing up.
  1. State clearly the hypotheses that should be used and any necessary assumption that needs to be made. [2]
  2. Carry out the test at the 1\% level of significance. [7]
Edexcel S4 2012 June Q2
16 marks Challenging +1.2
A biologist investigating the shell size of turtles takes random samples of adult female and adult male turtles and records the length, \(x\) cm, of the shell. The results are summarised below.
Number in sampleSample mean \(\bar{x}\)\(\sum x^2\)
Female619.62308.01
Male1213.72262.57
You may assume that the samples come from independent normal distributions with the same variance. The biologist claims that the mean shell length of adult female turtles is 5 cm longer than the mean shell length of adult male turtles.
  1. Test the biologist's claim at the 5\% level of significance. [10]
  2. Given that the true values for the variance of the population of adult male turtles and adult female turtles are both 0.9 cm\(^2\),
    1. show that when samples of size 6 and 12 are used with a 5\% level of significance, the biologist's claim will be accepted if \(4.07 < \bar{X}_F - \bar{X}_M < 5.93\) where \(\bar{X}_F\) and \(\bar{X}_M\) are the mean shell lengths of females and males respectively.
    2. Hence find the probability of a type II error for this test if in fact the true mean shell length of adult female turtles is 6 cm more than the mean shell length of adult male turtles. [6]
Edexcel S4 2012 June Q3
5 marks Standard +0.3
The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders' merchant is 36 mm\(^2\). The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders' merchant is 225 mm\(^2\). Test at the 10\% significance level whether or not there is evidence that the lengths of paving slabs sold by these builders' merchants differ in variability. State your hypotheses clearly. (You may assume the lengths of paving slabs are normally distributed.) [5]
Edexcel S4 2012 June Q4
16 marks Standard +0.3
A newspaper runs a daily Sudoku. A random sample of 10 people took the following times, in minutes, to complete the Sudoku. 5.0 \quad 4.5 \quad 4.7 \quad 5.3 \quad 5.2 \quad 4.1 \quad 5.3 \quad 4.8 \quad 5.5 \quad 4.6 Given that the times to complete the Sudoku follow a normal distribution,
  1. calculate a 95\% confidence interval for
    1. the mean,
    2. the variance,
    of the times taken by people to complete the Sudoku. [13] The newspaper requires the average time needed to complete the Sudoku to be 5 minutes with a standard deviation of 0.7 minutes.
  2. Comment on whether or not the Sudoku meets this requirement. Give a reason for your answer. [3]
Edexcel S4 2012 June Q5
13 marks Standard +0.3
Boxes of chocolates manufactured by Philippe have a mean weight of \(\mu\) grams and a standard deviation of \(\sigma\) grams. A random sample of 25 of these boxes are weighed. Using this sample, the unbiased estimate of \(\mu\) is 455 and the unbiased estimate of \(\sigma^2\) is 55.
  1. Test, at the 5\% level of significance, whether or not \(\sigma\) is greater than 6. State your hypotheses clearly. [6]
  2. Test, at the 5\% level of significance, whether or not \(\mu\) is more than 450. [6]
  3. State an assumption you have made in order to carry out the above tests. [1]
Edexcel S4 2012 June Q6
16 marks Standard +0.3
When a tree seed is planted the probability of it germinating is \(p\). A random sample of size \(n\) is taken and the number of tree seeds, \(X\), which germinate is recorded.
    1. Show that \(\hat{p}_1 = \frac{X}{n}\) is an unbiased estimator of \(p\).
    2. Find the variance of \(\hat{p}_1\). [4]
    A second sample of size \(m\) is taken and the number of tree seeds, \(Y\), which germinate is recorded. Given that \(\hat{p}_2 = \frac{Y}{m}\) and that \(\hat{p}_3 = a(3\hat{p}_1 + 2\hat{p}_2)\) is an unbiased estimator of \(p\),
  2. show that
    1. \(a = \frac{1}{5}\),
    2. \(\text{Var}(\hat{p}_3) = \frac{p(1-p)}{25}\left(\frac{9}{n} + \frac{4}{m}\right)\). [6]
  3. Find the range of values of \(\frac{n}{m}\) for which $$\text{Var}(\hat{p}_3) < \text{Var}(\hat{p}_1) \text{ and } \text{Var}(\hat{p}_3) < \text{Var}(\hat{p}_2)$$ [3]
  4. Given that \(n = 20\) and \(m = 60\), explain which of \(\hat{p}_1\), \(\hat{p}_2\) or \(\hat{p}_3\) is the best estimator. [3]