Questions — CAIE S2 (717 questions)

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CAIE S2 2010 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{29ab8740-9fad-4596-b34f-c424b5464858-2_453_1258_251_447} Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function f is shown above.
  1. State the value of \(k\).
  2. Explain briefly what this graph tells you about the arrival times of trains.
CAIE S2 2010 June Q2
2 A random sample of \(n\) people were questioned about their internet use. 87 of them had a high-speed internet connection. A confidence interval for the population proportion having a high-speed internet connection is \(0.1129 < p < 0.1771\).
  1. Write down the mid-point of this confidence interval and hence find the value of \(n\).
  2. This interval is an \(\alpha \%\) confidence interval. Find \(\alpha\).
CAIE S2 2010 June Q3
3 Metal bolts are produced in large numbers and have lengths which are normally distributed with mean 2.62 cm and standard deviation 0.30 cm .
  1. Find the probability that a random sample of 45 bolts will have a mean length of more than 2.55 cm .
  2. The machine making these bolts is given an annual service. This may change the mean length of bolts produced but does not change the standard deviation. To test whether the mean has changed, a random sample of 30 bolts is taken and their lengths noted. The sample mean length is \(m \mathrm {~cm}\). Find the set of values of \(m\) which result in rejection at the \(10 \%\) significance level of the hypothesis that no change in the mean length has occurred.
CAIE S2 2010 June Q4
4 The weekly distance in kilometres driven by Mr Parry has a normal distribution with mean 512 and standard deviation 62. Independently, the weekly distance in kilometres driven by Mrs Parry has a normal distribution with mean 89 and standard deviation 7.4.
  1. Find the probability that, in a randomly chosen week, Mr Parry drives more than 5 times as far as Mrs Parry.
  2. Find the mean and standard deviation of the total of the weekly distances in miles driven by Mr Parry and Mrs Parry. Use the approximation 8 kilometres \(= 5\) miles.
CAIE S2 2010 June Q5
5 The random variable \(T\) denotes the time in seconds for which a firework burns before exploding. The probability density function of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} k \mathrm { e } ^ { 0.2 t } & 0 \leqslant t \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 5 ( \mathrm { e } - 1 ) }\).
  2. Sketch the probability density function.
  3. \(80 \%\) of fireworks burn for longer than a certain time before they explode. Find this time.
CAIE S2 2010 June Q6
6 In restaurant \(A\) an average of 2.2\% of tablecloths are stained and, independently, in restaurant \(B\) an average of 5.8\% of tablecloths are stained.
  1. Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson approximation to find the probability that a total of more than 2 tablecloths are stained.
  2. Random samples of \(n\) tablecloths are taken from each restaurant. The probability that at least one tablecloth is stained is greater than 0.99 . Find the least possible value of \(n\).
CAIE S2 2010 June Q7
7 A hospital patient's white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient's white blood cell count is made, and is used to test at the \(10 \%\) significance level whether the mean white blood cell count has decreased.
  1. State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.
  2. Given that the measured value of the white blood cell count after the treatment is 2 , carry out the test.
  3. Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.
CAIE S2 2010 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{f73b303b-eb56-4c35-aa3e-388e3a3e8acd-2_453_1258_251_447} Fred arrives at random times on a station platform. The times in minutes he has to wait for the next train are modelled by the continuous random variable for which the probability density function f is shown above.
  1. State the value of \(k\).
  2. Explain briefly what this graph tells you about the arrival times of trains.
CAIE S2 2010 June Q1
1 At the 2009 election, \(\frac { 1 } { 3 }\) of the voters in Chington voted for the Citizens Party. One year later, a researcher questioned 20 randomly selected voters in Chington. Exactly 3 of these 20 voters said that if there were an election next week they would vote for the Citizens Party. Test at the \(2.5 \%\) significance level whether there is evidence of a decrease in support for the Citizens Party in Chington, since the 2009 election.
CAIE S2 2010 June Q2
2 Dipak carries out a test, at the \(10 \%\) significance level, using a normal distribution. The null hypothesis is \(\mu = 35\) and the alternative hypothesis is \(\mu \neq 35\).
  1. Is this a one-tail or a two-tail test? State briefly how you can tell. Dipak finds that the value of the test statistic is \(z = - 1.750\).
  2. Explain what conclusion he should draw.
  3. This result is significant at the \(\alpha \%\) level. Find the smallest possible value of \(\alpha\), correct to the nearest whole number.
CAIE S2 2010 June Q3
3 The weight, in grams, of a certain type of apple is modelled by the random variable \(X\) with mean 62 and standard deviation 8.2. A random sample of 50 apples is selected, and the mean weight in grams, \(\bar { X }\), is found.
  1. Describe fully the distribution of \(\bar { X }\).
  2. Find \(\mathrm { P } ( \bar { X } > 64 )\).
CAIE S2 2010 June Q4
4 At a power plant, the number of breakdowns per year has a Poisson distribution. In the past the mean number of breakdowns per year has been 4.8. Following some repairs, the management carry out a hypothesis test at the 5\% significance level to determine whether this mean has decreased. If there is at most 1 breakdown in the following year, they will conclude that the mean has decreased.
  1. State what is meant by a Type I error in this context.
  2. Find the probability of a Type I error.
  3. Find the probability of a Type II error if the mean is now 0.9 breakdowns per year.
CAIE S2 2010 June Q5
5 The time, in minutes, taken by volunteers to complete a task is modelled by the random variable \(X\) with probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { 4 } } & x \geqslant 1
0 & \text { otherwise. } \end{cases}$$
  1. Show that \(k = 3\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S2 2010 June Q6
6 Yu Ming travels to work and returns home once each day. The times, in minutes, that he takes to travel to work and to return home are represented by the independent random variables \(W\) and \(H\) with distributions \(\mathrm { N } \left( 22.4,4.8 ^ { 2 } \right)\) and \(\mathrm { N } \left( 20.3,5.2 ^ { 2 } \right)\) respectively.
  1. Find the probability that Yu Ming's total travelling time during a 5-day period is greater than 180 minutes.
  2. Find the probability that, on a particular day, Yu Ming takes longer to return home than he takes to travel to work.
CAIE S2 2010 June Q7
7 A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 4.2 )\).
  1. State two assumptions required for the Poisson model to be valid.
  2. Find the probability that
    (a) at least 1 patient will arrive in a 15-minute period,
    (b) fewer than 4 patients will arrive in a 10-minute period.
  3. The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough vaccine for 370 patients. Use a suitable approximation to find the probability that this will not be enough vaccine for that week.
CAIE S2 2011 June Q1
1 A hotel kitchen has two dish-washing machines. The numbers of breakdowns per year by the two machines have independent Poisson distributions with means 0.7 and 1.0 . Find the probability that the total number of breakdowns by the two machines during the next two years will be less than 3 .
CAIE S2 2011 June Q2
2 In a random sample of 70 bars of Luxcleanse soap, 18 were found to be undersized.
  1. Calculate an approximate \(90 \%\) confidence interval for the proportion of all bars of Luxcleanse soap that are undersized.
  2. Give a reason why your interval is only approximate.
CAIE S2 2011 June Q3
3 At an election in 2010, 15\% of voters in Bratfield voted for the Renewal Party. One year later, a researcher asked 30 randomly selected voters in Bratfield whether they would vote for the Renewal Party if there were an election next week. 2 of these 30 voters said that they would.
  1. Use a binomial distribution to test, at the \(4 \%\) significance level, the null hypothesis that there has been no change in the support for the Renewal Party in Bratfield against the alternative hypothesis that there has been a decrease in support since the 2010 election.
  2. (a) Explain why the conclusion in part (i) cannot involve a Type I error.
    (b) State the circumstances in which the conclusion in part (i) would involve a Type II error.
CAIE S2 2011 June Q4
4 On average, 1 in 2500 people have a particular gene.
  1. Use a suitable approximation to find the probability that, in a random sample of 10000 people, more than 3 people have this gene.
  2. The probability that, in a random sample of \(n\) people, none of them has the gene is less than 0.01 . Find the smallest possible value of \(n\).
CAIE S2 2011 June Q5
5 Each drink from a coffee machine contains \(X \mathrm {~cm} ^ { 3 }\) of coffee and \(Y \mathrm {~cm} ^ { 3 }\) of milk, where \(X\) and \(Y\) are independent variables with \(X \sim \mathrm {~N} \left( 184,15 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 50,8 ^ { 2 } \right)\). If the total volume of the drink is less than \(200 \mathrm {~cm} ^ { 3 }\) the customer receives the drink without charge.
  1. Find the percentage of drinks which customers receive without charge.
  2. Find the probability that, in a randomly chosen drink, the volume of coffee is more than 4 times the volume of milk.
CAIE S2 2011 June Q6
6 The distance travelled, in kilometres, by a Grippo brake pad before it needs to be replaced is modelled by \(10000 X\), where \(X\) is a random variable having the probability density function $$f ( x ) = \begin{cases} - k \left( x ^ { 2 } - 5 x + 6 \right) & 2 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ The graph of \(y = \mathrm { f } ( x )\) is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c1dcf0f5-e971-4afd-81ca-4d860732825c-3_439_1100_580_520}
  1. Show that \(k = 6\).
  2. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
  3. Sami fits four new Grippo brake pads on his car. Find the probability that at least one of these brake pads will need to be replaced after travelling less than 22000 km .
CAIE S2 2011 June Q7
7 Previous records have shown that the number of cars entering Bampor on any day has mean 352 and variance 121.
  1. Find the probability that the mean number of cars entering Bampor during a random sample of 200 days is more than 354 .
  2. State, with a reason, whether it was necessary to assume that the number of cars entering Bampor on any day has a normal distribution in order to find the probability in part (i).
  3. It is thought that the population mean may recently have changed. The number of cars entering Bampor during the day was recorded for each of a random sample of 50 days and the sample mean was found to be 356 . Assuming that the variance is unchanged, test at the \(5 \%\) significance level whether the population mean is still 352 .
CAIE S2 2012 June Q1
1 The number of new enquiries per day at an office has a Poisson distribution. In the past the mean has been 3 . Following a change of staff, the manager wishes to test, at the \(5 \%\) significance level, whether the mean has increased.
  1. State the null and alternative hypotheses for this test. The manager notes the number, \(N\), of new enquiries during a certain 6 -day period. She finds that \(N = 25\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \geqslant 25 ) = 0.0683\).
  2. What conclusion should she draw?
CAIE S2 2012 June Q2
2 A population has mean 7 and standard deviation 3. A random sample of size \(n\) is chosen from this population.
  1. Write down the mean and standard deviation of the distribution of the sample mean.
  2. Under what circumstances does the sample mean have
    (a) a normal distribution,
    (b) an approximately normal distribution?
CAIE S2 2012 June Q3
3 In a sample of 50 students at Batlin college, 18 support the football club Real Madrid.
  1. Calculate an approximate \(98 \%\) confidence interval for the proportion of students at Batlin college who support Real Madrid.
  2. Give one condition for this to be a reliable result.