Questions — CAIE S2 (737 questions)

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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
5 marks 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
8 marks 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
10 marks 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
10 marks 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
12 marks Challenging +1.2
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.
    1. On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
    2. Find the probability that there are more than 1250 worms on a one kilometre length of this path.
CAIE S2 2008 November Q1
4 marks 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
5 marks 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
5 marks 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
7 marks 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
8 marks 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
9 marks 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.
CAIE S2 2008 November Q7
12 marks Standard +0.3
7 The time in hours taken for clothes to dry can be modelled by the continuous random variable with probability density function given by $$f ( t ) = \begin{cases} k \sqrt { } t & 1 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 14 }\).
  2. Find the mean time taken for clothes to dry.
  3. Find the median time taken for clothes to dry.
  4. Find the probability that the time taken for clothes to dry is between the mean time and the median time.
CAIE S2 2009 November Q2
4 marks Standard +0.8
2 The lengths of sewing needles in travel sewing kits are distributed normally with mean \(\mu \mathrm { mm }\) and standard deviation 1.5 mm . A random sample of \(n\) needles is taken. Find the smallest value of \(n\) such that the width of a \(95 \%\) confidence interval for the population mean is at most 1 mm .
CAIE S2 2009 November Q3
7 marks Standard +0.3
3 The weights of pebbles on a beach are normally distributed with mean 48.5 grams and standard deviation 12.4 grams.
  1. Find the probability that the mean weight of a random sample of 5 pebbles is greater than 51 grams.
  2. The probability that the mean weight of a random sample of \(n\) pebbles is less than 51.6 grams is 0.9332 . Find the value of \(n\).
CAIE S2 2009 November Q4
8 marks Challenging +1.2
4 The number of severe floods per year in a certain country over the last 100 years has followed a Poisson distribution with mean 1.8. Scientists suspect that global warming has now increased the mean. A hypothesis test, at the \(5 \%\) significance level, is to be carried out to test this suspicion. The number of severe floods, \(X\), that occur next year will be used for the test.
  1. Show that the rejection region for the test is \(X > 4\).
  2. Find the probability of making a Type II error if the mean number of severe floods is now actually 2.3.
CAIE S2 2009 November Q5
9 marks Standard +0.3
5 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \sqrt { } 2\).
  2. Find \(\mathrm { P } ( X > 0.4 )\).
  3. Find the upper quartile of \(X\).
  4. Find the probability that exactly 3 out of 5 random observations of \(X\) have values greater than the upper quartile.
CAIE S2 2009 November Q6
10 marks Standard +0.3
6 Photographers often need to take many photographs of families until they find a photograph which everyone in the family likes. The number of photographs taken until obtaining one which everybody likes has mean 15.2. A new photographer claims that she can obtain a photograph which everybody likes with fewer photographs taken. To test at the \(10 \%\) level of significance whether this claim is justified, the numbers of photographs, \(x\), taken by the new photographer with a random sample of 60 families are recorded. The results are summarised by \(\Sigma x = 890\) and \(\Sigma x ^ { 2 } = 13780\).
  1. Calculate unbiased estimates of the population mean and variance of the number of photographs taken by the new photographer.
  2. State null and alternative hypotheses for the test, and state also the probability that the test results in a Type I error. Say what a Type I error means in the context of the question.
  3. Carry out the test.
CAIE S2 2009 November Q7
9 marks Standard +0.8
7 The volume of liquid in cans of cola is normally distributed with mean 330 millilitres and standard deviation 5.2 millilitres. The volume of liquid in bottles of tonic water is normally distributed with mean 500 millilitres and standard deviation 7.1 millilitres.
  1. Find the probability that 3 randomly chosen cans of cola contain less liquid than 2 randomly chosen bottles of tonic water.
  2. A new drink is made by mixing the contents of 2 cans of cola with half a bottle of tonic water. Find the probability that the volume of the new drink is more than 900 millilitres.
CAIE S2 2009 November Q1
4 marks Easy -1.2
1 There are 18 people in Millie's class. To choose a person at random she numbers the people in the class from 1 to 18 and presses the random number button on her calculator to obtain a 3-digit decimal. Millie then multiplies the first digit in this decimal by two and chooses the person corresponding to this new number. Decimals in which the first digit is zero are ignored.
  1. Give a reason why this is not a satisfactory method of choosing a person. Millie obtained a random sample of 5 people of her own age by a satisfactory sampling method and found that their heights in metres were \(1.66,1.68,1.54,1.65\) and 1.57 . Heights are known to be normally distributed with variance \(0.0052 \mathrm {~m} ^ { 2 }\).
  2. Find a \(98 \%\) confidence interval for the mean height of people of Millie's age.
CAIE S2 2009 November Q2
6 marks Standard +0.3
2 A computer user finds that unwanted emails arrive randomly at a uniform average rate of 1.27 per hour.
  1. Find the probability that more than 1 unwanted email arrives in a period of 5 hours.
  2. Find the probability that more than 850 unwanted emails arrive in a period of 700 hours.
CAIE S2 2009 November Q3
7 marks Standard +0.3
3 An airline knows that some people who have bought tickets may not arrive for the flight. The airline therefore sells more tickets than the number of seats that are available. For one flight there are 210 seats available and 213 people have bought tickets. The probability of any person who has bought a ticket not arriving for the flight is \(\frac { 1 } { 50 }\).
  1. By considering the number of people who do not arrive for the flight, use a suitable approximation to calculate the probability that more people will arrive than there are seats available. Independently, on another flight for which 135 people have bought tickets, the probability of any person not arriving is \(\frac { 1 } { 75 }\).
  2. Calculate the probability that, for both these flights, the total number of people who do not arrive is 5 .
CAIE S2 2009 November Q4
7 marks Challenging +1.2
4 It is not known whether a certain coin is fair or biased. In order to perform a hypothesis test, Raj tosses the coin 10 times and counts the number of heads obtained. The probability of obtaining a head on any throw is denoted by \(p\).
  1. The null hypothesis is \(p = 0.5\). Find the acceptance region for the test, given that the probability of a Type I error is to be at most 0.1 .
  2. Calculate the probability of a Type II error in this test if the actual value of \(p\) is 0.7 .
CAIE S2 2009 November Q5
8 marks Standard +0.8
5 The masses of packets of cornflakes are normally distributed with standard deviation 11 g . A random sample of 20 packets was weighed and found to have a mean mass of 746 g .
  1. Test at the \(4 \%\) significance level whether there is enough evidence to conclude that the population mean mass is less than 750 g .
  2. Given that the population mean mass actually is 750 g , find the smallest possible sample size, \(n\), for which it is at least \(97 \%\) certain that the mean mass of the sample exceeds 745 g .
CAIE S2 2009 November Q6
8 marks Standard +0.3
6 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 3 } x ( k - x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that the value of \(k\) is \(\frac { 32 } { 9 }\).
  2. Find \(\mathrm { E } ( X )\).
  3. Is the median less than or greater than 1.5? Justify your answer numerically.