Questions S2 (1597 questions)

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CAIE S2 2005 November Q2
Moderate -0.3
2 A manufacturer claims that \(20 \%\) of sugar-coated chocolate beans are red. George suspects that this percentage is actually less than \(20 \%\) and so he takes a random sample of 15 chocolate beans and performs a hypothesis test with the null hypothesis \(p = 0.2\) against the alternative hypothesis \(p < 0.2\). He decides to reject the null hypothesis in favour of the alternative hypothesis if there are 0 or 1 red beans in the sample.
  1. With reference to this situation, explain what is meant by a Type I error.
  2. Find the probability of a Type I error in George's test.
CAIE S2 2005 November Q3
Standard +0.3
3 Flies stick to wet paint at random points. The average number of flies is 2 per square metre. A wall with area \(22 \mathrm {~m} ^ { 2 }\) is painted with a new type of paint which the manufacturer claims is fly-repellent. It is found that 27 flies stick to this wall. Use a suitable approximation to test the manufacturer's claim at the \(1 \%\) significance level. Take the null hypothesis to be \(\mu = 44\), where \(\mu\) is the population mean.
CAIE S2 2005 November Q4
Moderate -0.3
4
  1. Give a reason why, in carrying out a statistical investigation, a sample rather than a complete population may be used.
  2. Rose wishes to investigate whether men in her town have a different life-span from the national average of 71.2 years. She looks at government records for her town and takes a random sample of the ages of 110 men who have died recently. Their mean age in years was 69.3 and the unbiased estimate of the population variance was 65.61.
    (a) Calculate a \(90 \%\) confidence interval for the population mean and explain what you understand by this confidence interval.
    (b) State with a reason what conclusion about the life-span of men in her town Rose could draw from this confidence interval.
CAIE S2 2005 November Q5
Moderate -0.8
5 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 3 } x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that the value of \(a\) is \(\frac { 1 } { 2 }\).
  2. Find \(\mathrm { P } ( X > 1.8 )\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2005 November Q6
Standard +0.3
6 A shopkeeper sells electric fans. The demand for fans follows a Poisson distribution with mean 3.2 per week.
  1. Find the probability that the demand is exactly 2 fans in any one week.
  2. The shopkeeper has 4 fans in his shop at the beginning of a week. Find the probability that this will not be enough to satisfy the demand for fans in that week.
  3. Given instead that he has \(n\) fans in his shop at the beginning of a week, find, by trial and error, the least value of \(n\) for which the probability of his not being able to satisfy the demand for fans in that week is less than 0.05 .
CAIE S2 2005 November Q7
Challenging +1.2
7 A journey in a certain car consists of two stages with a stop for filling up with fuel after the first stage. The length of time, \(T\) minutes, taken for each stage has a normal distribution with mean 74 and standard deviation 7.3. The length of time, \(F\) minutes, it takes to fill up with fuel has a normal distribution with mean 5 and standard deviation 1.7. The length of time it takes to pay for the fuel is exactly 4 minutes. The variables \(T\) and \(F\) are independent and the times for the two stages are independent of each other.
  1. Find the probability that the total time for the journey is less than 154 minutes.
  2. A second car has a fuel tank with exactly twice the capacity of the first car. Find the mean and variance of this car's fuel fill-up time.
  3. This second car's time for each stage of the journey follows a normal distribution with mean 69 minutes and standard deviation 5.2 minutes. The length of time it takes to pay for the fuel for this car is also exactly 4 minutes. Find the probability that the total time for the journey taken by the first car is more than the total time taken by the second car.
CAIE S2 2006 November Q1
Moderate -0.8
1 The time taken for Samuel to drive home from work is distributed with mean 46 minutes. Samuel discovers a different route and decides to test at the \(5 \%\) level whether the mean time has changed. He tries this route on a large number of different days chosen randomly and calculates the mean time.
  1. State the null and alternative hypotheses for this test.
  2. Samuel calculates the value of his test statistic \(z\) to be - 1.729 . What conclusion can he draw?
CAIE S2 2006 November Q2
Easy -1.2
2
  1. Write down the mean and variance of the distribution of the means of random samples of size \(n\) taken from a very large population having mean \(\mu\) and variance \(\sigma ^ { 2 }\).
  2. What, if anything, can you say about the distribution of sample means
    (a) if \(n\) is large,
    (b) if \(n\) is small?
CAIE S2 2006 November Q3
Moderate -0.8
3 A survey was conducted to find the proportion of people owning DVD players. It was found that 203 out of a random sample of 278 people owned a DVD player.
  1. Calculate a \(97 \%\) confidence interval for the true proportion of people who own a DVD player. A second survey to find the proportion of people owning DVD players was conducted at 10 o'clock on a Thursday morning in a shopping centre.
  2. Give one reason why this is not a satisfactory sample.
CAIE S2 2006 November Q4
Standard +0.3
4 In summer, wasps' nests occur randomly in the south of England at an average rate of 3 nests for every 500 houses.
  1. Find the probability that two villages in the south of England, with 600 houses and 700 houses, have a total of exactly 3 wasps' nests.
  2. Use a suitable approximation to estimate the probability of there being fewer than 369 wasps' nests in a town with 64000 houses.
CAIE S2 2006 November Q5
Standard +0.3
5 Climbing ropes produced by a manufacturer have breaking strengths which are normally distributed with mean 160 kg and standard deviation 11.3 kg . A group of climbers have weights which are normally distributed with mean 66.3 kg and standard deviation 7.1 kg .
  1. Find the probability that a rope chosen randomly will break under the combined weight of 2 climbers chosen randomly. Each climber carries, in a rucksack, equipment amounting to half his own weight.
  2. Find the mean and variance of the combined weight of a climber and his rucksack.
  3. Find the probability that the combined weight of a climber and his rucksack is greater than 87 kg .
CAIE S2 2006 November Q6
Standard +0.8
6 Pieces of metal discovered by people using metal detectors are found randomly in fields in a certain area at an average rate of 0.8 pieces per hectare. People using metal detectors in this area have a theory that ploughing the fields increases the average number of pieces of metal found per hectare. After ploughing, they tested this theory and found that a randomly chosen field of area 3 hectares yielded 5 pieces of metal.
  1. Carry out the test at the \(10 \%\) level of significance.
  2. What would your conclusion have been if you had tested at the \(5 \%\) level of significance? Jack decides that he will reject the null hypothesis that the average number is 0.8 pieces per hectare if he finds 4 or more pieces of metal in another ploughed field of area 3 hectares.
  3. If the true mean after ploughing is 1.4 pieces per hectare, calculate the probability that Jack makes a Type II error.
CAIE S2 2006 November Q7
Standard +0.3
7 At a town centre car park the length of stay in hours is denoted by the random variable \(X\), which has probability density function given by $$f ( x ) = \begin{cases} k x ^ { - \frac { 3 } { 2 } } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Interpret the inequalities \(1 \leqslant x \leqslant 9\) in the definition of \(\mathrm { f } ( x )\) in the context of the question.
  2. Show that \(k = \frac { 3 } { 4 }\).
  3. Calculate the mean length of stay. The charge for a length of stay of \(x\) hours is \(\left( 1 - \mathrm { e } ^ { - x } \right)\) dollars.
  4. Find the length of stay for the charge to be at least 0.75 dollars
  5. Find the probability of the charge being at least 0.75 dollars.
CAIE S2 2007 November Q1
5 marks Moderate -0.3
1 Isaac claims that \(30 \%\) of cars in his town are red. His friend Hardip thinks that the proportion is less than \(30 \%\). The boys decided to test Isaac's claim at the \(5 \%\) significance level and found that 2 cars out of a random sample of 18 were red. Carry out the hypothesis test and state your conclusion. [5]
CAIE S2 2007 November Q2
Standard +0.8
2 In summer the growth rate of grass in a lawn has a normal distribution with mean 3.2 cm per week and standard deviation 1.4 cm per week. A new type of grass is introduced which the manufacturer claims has a slower growth rate. A hypothesis test of this claim at the \(5 \%\) significance level was carried out using a random sample of 10 lawns that had the new grass. It may be assumed that the growth rate of the new grass has a normal distribution with standard deviation 1.4 cm per week.
  1. Find the rejection region for the test.
  2. The probability of making a Type II error when the actual value of the mean growth rate of the new grass is \(m \mathrm {~cm}\) per week is less than 0.5 . Use your answer to part (i) to write down an inequality for \(m\).
CAIE S2 2007 November Q3
Moderate -0.8
3
  1. Explain what is meant by the term 'random sample'. In a random sample of 350 food shops it was found that 130 of them had Special Offers.
  2. Calculate an approximate \(95 \%\) confidence interval for the proportion of all food shops with Special Offers.
  3. Estimate the size of a random sample required for an approximate \(95 \%\) confidence interval for this proportion to have a width of 0.04 .
CAIE S2 2007 November Q4
Standard +0.3
4 The cost of electricity for a month in a certain town under scheme \(A\) consists of a fixed charge of 600 cents together with a charge of 5.52 cents per unit of electricity used. Stella uses scheme \(A\). The number of units she uses in a month is normally distributed with mean 500 and variance 50.41.
  1. Find the mean and variance of the total cost of Stella's electricity in a randomly chosen month. Under scheme \(B\) there is no fixed charge and the cost in cents for a month is normally distributed with mean 6600 and variance 421. Derek uses scheme \(B\).
  2. Find the probability that, in a randomly chosen month, Derek spends more than twice as much as Stella spends.
CAIE S2 2007 November Q5
Moderate -0.5
5 The length, \(X \mathrm {~cm}\), of a piece of wooden planking is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(b\) is a positive constant.
  1. Find the mean and variance of \(X\) in terms of \(b\). The lengths of a random sample of 100 pieces were measured and it was found that \(\Sigma x = 950\).
  2. Show that the value of \(b\) estimated from this information is 19 . Using this value of \(b\),
  3. find the probability that the length of a randomly chosen piece is greater than 11 cm ,
  4. find the probability that the mean length of a random sample of 336 pieces is less than 9 cm .
CAIE S2 2007 November Q6
6 The random variable \(X\) denotes the number of worms on a one metre length of a country path after heavy rain. It is given that \(X\) has a Poisson distribution.
  1. For one particular path, the probability that \(X = 2\) is three times the probability that \(X = 4\). Find the probability that there are more than 3 worms on a 3.5 metre length of this path.
  2. For another path the mean of \(X\) is 1.3.
    (a) On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
    (b) Find the probability that there are more than 1250 worms on a one kilometre length of this path.
CAIE S2 2008 November Q1
Easy -1.2
1 Alan wishes to choose one child at random from the eleven children in his music class. The children are numbered \(2,3,4\), and so on, up to 12 . Alan then throws two fair dice, each numbered from 1 to 6 , and chooses the child whose number is the sum of the scores on the two dice.
  1. Explain why this is an unsatisfactory method of choosing a child.
  2. Describe briefly a satisfactory method of choosing a child.
CAIE S2 2008 November Q2
Moderate -0.3
2 The times taken for the pupils in Ming's year group to do their English homework have a normal distribution with standard deviation 15.7 minutes. A teacher estimates that the mean time is 42 minutes. The times taken by a random sample of 3 students from the year group were 27, 35 and 43 minutes. Carry out a hypothesis test at the \(10 \%\) significance level to determine whether the teacher's estimate for the mean should be accepted, stating the null and alternative hypotheses.
CAIE S2 2008 November Q3
Standard +0.8
3 Weights of garden tables are normally distributed with mean 36 kg and standard deviation 1.6 kg . Weights of garden chairs are normally distributed with mean 7.3 kg and standard deviation 0.4 kg . Find the probability that the total weight of 2 randomly chosen tables is more than the total weight of 10 randomly chosen chairs.
CAIE S2 2008 November Q4
Standard +0.3
4 Diameters of golf balls are known to be normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). A random sample of 130 golf balls was taken and the diameters, \(x \mathrm {~cm}\), were measured. The results are summarised by \(\Sigma x = 555.1\) and \(\Sigma x ^ { 2 } = 2371.30\).
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Calculate a \(97 \%\) confidence interval for \(\mu\).
  3. 300 random samples of 130 balls are taken and a \(97 \%\) confidence interval is calculated for each sample. How many of these intervals would you expect not to contain \(\mu\) ?
CAIE S2 2008 November Q5
Moderate -0.3
5 Every month Susan enters a particular lottery. The lottery company states that the probability, \(p\), of winning a prize is 0.0017 each month. Susan thinks that the probability of winning is higher than this, and carries out a test based on her 12 lottery results in a one-year period. She accepts the null hypothesis \(p = 0.0017\) if she has no wins in the year and accepts the alternative hypothesis \(p > 0.0017\) if she wins a prize in at least one of the 12 months.
  1. Find the probability of the test resulting in a Type I error.
  2. If in fact the probability of winning a prize each month is 0.0024 , find the probability of the test resulting in a Type II error.
  3. Use a suitable approximation, with \(p = 0.0024\), to find the probability that in a period of 10 years Susan wins a prize exactly twice.
CAIE S2 2008 November Q6
Standard +0.3
6 In their football matches, Rovers score goals independently and at random times. Their average rate of scoring is 2.3 goals per match.
  1. State the expected number of goals that Rovers will score in the first half of a match.
  2. Find the probability that Rovers will not score any goals in the first half of a match but will score one or more goals in the second half of the match.
  3. Football matches last for 90 minutes. In a particular match, Rovers score one goal in the first 30 minutes. Find the probability that they will score at least one further goal in the remaining 60 minutes. Independently of the number of goals scored by Rovers, the number of goals scored per football match by United has a Poisson distribution with mean 1.8.
  4. Find the probability that a total of at least 3 goals will be scored in a particular match when Rovers play United.