Questions — CAIE S2 (717 questions)

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CAIE S2 2014 November Q4
4 marks
4 In a survey a random sample of 150 households in Nantville were asked to fill in a questionnaire about household budgeting.
  1. The results showed that 33 households owned more than one car. Find an approximate \(99 \%\) confidence interval for the proportion of all households in Nantville with more than one car. [4]
  2. The results also included the weekly expenditure on food, \(x\) dollars, of the households. These were summarised as follows. $$n = 150 \quad \Sigma x = 19035 \quad \Sigma x ^ { 2 } = 4054716$$ Find unbiased estimates of the mean and variance of the weekly expenditure on food of all households in Nantville.
  3. The government has a list of all the households in Nantville numbered from 1 to 9526. Describe briefly how to use random numbers to select a sample of 150 households from this list.
CAIE S2 2014 November Q5
5 The number of hours that Mrs Hughes spends on her business in a week is normally distributed with mean \(\mu\) and standard deviation 4.8. In the past the value of \(\mu\) has been 49.5.
  1. Assuming that \(\mu\) is still equal to 49.5 , find the probability that in a random sample of 40 weeks the mean time spent on her business in a week is more than 50.3 hours. Following a change in her arrangements, Mrs Hughes wishes to test whether \(\mu\) has decreased. She chooses a random sample of 40 weeks and notes that the total number of hours she spent on her business during these weeks is 1920.
  2. (a) Explain why a one-tail test is appropriate.
    (b) Carry out the test at the 6\% significance level.
    (c) Explain whether it was necessary to use the Central Limit theorem in part (ii) (b).
CAIE S2 2014 November Q6
6 The number of accidents on a certain road has a Poisson distribution with mean 3.1 per 12-week period.
  1. Find the probability that there will be exactly 4 accidents during an 18-week period. Following the building of a new junction on this road, an officer wishes to determine whether the number of accidents per week has decreased. He chooses 15 weeks at random and notes the number of accidents. If there are fewer than 3 accidents altogether he will conclude that the number of accidents per week has decreased. He assumes that a Poisson distribution still applies.
  2. Find the probability of a Type I error.
  3. Given that the mean number of accidents per week is now 0.1 , find the probability of a Type II error.
  4. Given that there were 2 accidents during the 15 weeks, explain why it is impossible for the officer to make a Type II error.
CAIE S2 2014 November Q3
3
  1. The time for which Lucy has to wait at a certain traffic light each day is \(T\) minutes, where \(T\) has probability density function given by $$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2
    0 & \text { otherwise } \end{cases}$$ Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.

  2. \includegraphics[max width=\textwidth, alt={}, center]{c08d3228-430e-4158-9362-1655deb1feb7-2_405_791_1471_715} The diagram shows the graph of the probability density function, g , of a random variable \(X\). The graph of g is a semicircle with centre \(( 0,0 )\) and radius \(a\). Elsewhere \(\mathrm { g } ( x ) = 0\).
    1. Find the value of \(a\).
    2. State the value of \(\mathrm { E } ( X )\).
    3. Given that \(\mathrm { P } ( X < - c ) = 0.2\), find \(\mathrm { P } ( X < c )\).
CAIE S2 2014 November Q1
1 A researcher wishes to investigate whether the mean height of a certain type of plant in one region is different from the mean height of this type of plant everywhere else. He takes a large random sample of plants from the region and finds the sample mean. He calculates the value of the test statistic, \(z\), and finds that \(z = 1.91\).
  1. Explain briefly why the researcher should use a two-tail test.
  2. Carry out the test at the \(4 \%\) significance level.
CAIE S2 2014 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{323cf83a-e23b-494e-a911-856d8f1c92fd-2_483_791_708_676} The diagram shows the graph of the probability density function, f , of a random variable \(X\).
  1. Find the value of the constant \(c\).
  2. Find the value of \(a\) such that \(\mathrm { P } ( a < X < 1 ) = 0.1\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S2 2014 November Q3
3 The times, in minutes, taken by people to complete a walk are normally distributed with mean \(\mu\). The times, \(t\) minutes, for a random sample of 80 people were summarised as follows. $$\Sigma t = 7220 \quad \Sigma t ^ { 2 } = 656060$$
  1. Calculate a \(97 \%\) confidence interval for \(\mu\).
  2. Explain whether it was necessary to use the Central Limit theorem in part (i).
CAIE S2 2014 November Q4
4 The masses, in grams, of tomatoes of type \(A\) and type \(B\) have the distributions \(\mathrm { N } \left( 125,30 ^ { 2 } \right)\) and \(\mathrm { N } \left( 130,32 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total mass of 4 randomly chosen tomatoes of type \(A\) and 6 randomly chosen tomatoes of type \(B\) is less than 1.5 kg .
  2. Find the probability that a randomly chosen tomato of type \(A\) has a mass that is at least \(90 \%\) of the mass of a randomly chosen tomato of type \(B\).
CAIE S2 2014 November Q5
5 It is known that when seeds of a certain type are planted, on average \(10 \%\) of the resulting plants reach a height of 1 metre. A gardener wishes to investigate whether a new fertiliser will increase this proportion. He plants a random sample of 18 seeds of this type, using the fertiliser, and notes how many of the resulting plants reach a height of 1 metre.
  1. In fact 4 of the 18 plants reach a height of 1 metre. Carry out a hypothesis test at the \(8 \%\) significance level.
  2. Explain which of the errors, Type I or Type II, might have been made in part (i). Later, the gardener plants another random sample of 18 seeds of this type, using the fertiliser, and again carries out a hypothesis test at the \(8 \%\) significance level.
  3. Find the probability of a Type I error.
CAIE S2 2014 November Q6
6 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5 -minute period. Find the probability of
  1. exactly 4 calls in an 8 -minute period,
  2. at least 3 calls in a 3-minute period. The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
  3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.
CAIE S2 2015 November Q1
1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .
CAIE S2 2015 November Q2
2 The mean and standard deviation of the time spent by people in a certain library are 29 minutes and 6 minutes respectively.
  1. Find the probability that the mean time spent in the library by a random sample of 120 people is more than 30 minutes.
  2. Explain whether it was necessary to assume that the time spent by people in the library is normally distributed in the solution to part (i).
CAIE S2 2015 November Q3
3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below. $$n = 60 \quad \Sigma x = 3420 \quad \Sigma x ^ { 2 } = 195200$$
  1. Calculate unbiased estimates of the population mean and variance.
  2. Calculate a \(98 \%\) confidence interval for the population mean.
CAIE S2 2015 November Q4
4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 2 } { 3 }\).
  2. Find the median of \(X\).
CAIE S2 2015 November Q5
5 On average, 1 in 2500 adults has a certain medical condition.
  1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).
CAIE S2 2015 November Q6
6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.
  1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
  2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man.
CAIE S2 2015 November Q7
7 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked.
  1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample. A suitable sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the 5\% significance level.
  2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
  3. State what is meant by a Type I error in this context.
  4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
  5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.
CAIE S2 2015 November Q1
1 It is known that the number, \(N\), of words contained in the leading article each day in a certain newspaper can be modelled by a normal distribution with mean 352 and variance 29. A researcher takes a random sample of 10 leading articles and finds the sample mean, \(\bar { N }\), of \(N\).
  1. State the distribution of \(\bar { N }\), giving the values of any parameters.
  2. Find \(\mathrm { P } ( \bar { N } > 354 )\).
CAIE S2 2015 November Q2
2 The number of calls received per 5-minute period at a large call centre has a Poisson distribution with mean \(\lambda\), where \(\lambda > 30\). If more than 55 calls are received in a 5 -minute period, the call centre is overloaded. It has been found that the probability of being overloaded during a randomly chosen 5 -minute period is 0.01 . Use the normal approximation to the Poisson distribution to obtain a quadratic equation in \(\sqrt { } \lambda\) and hence find the value of \(\lambda\).
CAIE S2 2015 November Q3
3 From a random sample of 65 people in a certain town, the proportion who own a bicycle was noted. From this result an \(\alpha \%\) confidence interval for the proportion, \(p\), of all people in the town who own a bicycle was calculated to be \(0.284 < p < 0.516\).
  1. Find the proportion of people in the sample who own a bicycle.
  2. Calculate the value of \(\alpha\) correct to 2 significant figures.
CAIE S2 2015 November Q4
4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \left( 4 - x ^ { 2 } \right) & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 32 }\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\) and hence write down the value of \(\mathrm { E } ( X )\).
  3. Find \(\mathrm { P } ( X < 1 )\).
CAIE S2 2015 November Q5
5
  1. Narika has a die which is known to be biased so that the probability of throwing a 6 on any throw is \(\frac { 1 } { 100 }\). She uses an approximating distribution to calculate the probability of obtaining no 6s in 450 throws. Find the percentage error in using the approximating distribution for this calculation.
  2. Johan claims that a certain six-sided die is biased so that it shows a 6 less often than it would if the die were fair. In order to test this claim, the die is thrown 25 times and it shows a 6 on only 2 throws. Test at the \(10 \%\) significance level whether Johan's claim is justified.
CAIE S2 2015 November Q6
6 Parcels arriving at a certain office have weights \(W \mathrm {~kg}\), where the random variable \(W\) has mean \(\mu\) and standard deviation 0.2 . The value of \(\mu\) used to be 2.60 , but there is a suspicion that this may no longer be true. In order to test at the 5\% significance level whether the value of \(\mu\) has increased, a random sample of 75 parcels is chosen. You may assume that the standard deviation of \(W\) is unchanged.
  1. The mean weight of the 75 parcels is found to be 2.64 kg . Carry out the test.
  2. Later another test of the same hypotheses at the \(5 \%\) significance level, with another random sample of 75 parcels, is carried out. Given that the value of \(\mu\) is now 2.68 , calculate the probability of a Type II error.
CAIE S2 2015 November Q7
7 The diameter, in cm, of pistons made in a certain factory is denoted by \(X\), where \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The diameters of a random sample of 100 pistons were measured, with the following results. $$n = 100 \quad \Sigma x = 208.7 \quad \Sigma x ^ { 2 } = 435.57$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\). The pistons are designed to fit into cylinders. The internal diameter, in cm , of the cylinders is denoted by \(Y\), where \(Y\) has an independent normal distribution with mean 2.12 and variance 0.000144 . A piston will not fit into a cylinder if \(Y - X < 0.01\).
  2. Using your answers to part (i), find the probability that a randomly chosen piston will not fit into a randomly chosen cylinder.
CAIE S2 2016 November Q1
1 The weights, in kilograms, of a random sample of eight 16-year old males are given below. $$\begin{array} { l l l l l l l l } 58.9 & 63.5 & 62.7 & 59.4 & 66.9 & 68.0 & 60.4 & 68.2 \end{array}$$ Find unbiased estimates of the population mean and variance of the weights of all 16 -year old males.