Questions S2 (1690 questions)

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CAIE S2 2016 June Q2
5 marks Easy -1.3
2 A researcher is investigating the lengths, in kilometres, of the journeys to work of the employees at a certain firm. She takes a random sample of 10 employees.
  1. State what is meant by 'random' in this context. The results of her sample are as follows. $$\begin{array} { l l l l l l l l l l } 1.5 & 2.0 & 3.6 & 5.9 & 4.8 & 8.7 & 3.5 & 2.9 & 4.1 & 3.0 \end{array}$$
  2. Find unbiased estimates of the population mean and variance.
  3. State what is meant by 'population' in this context.
CAIE S2 2016 June Q3
5 marks Moderate -0.8
3 Based on a random sample of 700 people living in a certain area, a confidence interval for the proportion, \(p\), of all people living in that area who had travelled abroad was found to be \(0.5672 < p < 0.6528\).
  1. Find the proportion of people in the sample who had travelled abroad.
  2. Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.
CAIE S2 2016 June Q4
6 marks Moderate -0.3
4 In the past, the time spent by customers in a certain shop had mean 12.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 13.5 minutes.
  1. Assuming that the standard deviation remains at 4.2 minutes, test at the \(5 \%\) significance level whether the mean time spent by customers in the shop has changed.
  2. Another random sample of 50 customers is chosen and a similar test at the \(5 \%\) significance level is carried out. State the probability of a Type I error.
CAIE S2 2016 June Q5
9 marks Standard +0.8
5 The thickness of books in a large library is normally distributed with mean 2.4 cm and standard deviation 0.3 cm .
  1. Find the probability that the total thickness of 6 randomly chosen books is more than 16 cm .
  2. Find the probability that the thickness of a book chosen at random is less than 1.1 times the thickness of a second book chosen at random.
CAIE S2 2016 June Q6
10 marks Moderate -0.3
6 In each turn of a game, a coin is pushed and slides across a table. The distance, \(X\) metres, travelled by the coin has probability density function given by $$f ( x ) = \begin{cases} k x ^ { 2 } ( 2 - x ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. State the greatest possible distance travelled by the coin in one turn.
  2. Show that \(k = \frac { 3 } { 4 }\).
  3. Find the mean distance travelled by the coin in one turn.
  4. Out of 400 turns, find the expected number of turns in which the distance travelled by the coin is less than 1 metre.
CAIE S2 2016 June Q7
10 marks Standard +0.3
7
  1. A large number of spoons and forks made in a factory are inspected. It is found that \(1 \%\) of the spoons and \(1.5 \%\) of the forks are defective. A random sample of 140 items, consisting of 80 spoons and 60 forks, is chosen. Use the Poisson approximation to the binomial distribution to find the probability that the sample contains
    1. at least 1 defective spoon and at least 1 defective fork,
    2. fewer than 3 defective items.
  2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\mathrm { P } ( X = 1 ) = p \quad \text { and } \quad \mathrm { P } ( X = 2 ) = 1.5 p$$ where \(p\) is a non-zero constant. Find the value of \(\lambda\) and hence find the value of \(p\).
CAIE S2 2016 June Q1
4 marks Moderate -0.5
1 The length of time, in minutes, taken by people to complete a task has mean 53.0 and standard deviation 6.2. Find the probability that the mean time taken to complete the task by a random sample of 50 people is more than 51 minutes.
CAIE S2 2016 June Q2
4 marks Moderate -0.3
2 Jacques is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Marie suspects that the true percentage is lower than \(90 \%\). She asks a random sample of 15 of Jacques' customers whether they are satisfied. She then performs a hypothesis test of the null hypothesis \(p = 0.9\) against the alternative hypothesis \(p < 0.9\), where \(p\) is the population proportion of customers who are satisfied. She decides to reject the null hypothesis if fewer than 12 customers are satisfied.
  1. In the context of the question, explain what is meant by a Type I error.
  2. Find the probability of a Type I error in Marie's test.
CAIE S2 2016 June Q3
5 marks Moderate -0.8
3
  1. Give a reason for using a sample rather than the whole population in carrying out a statistical investigation.
  2. Tennis balls of a certain brand are known to have a mean height of bounce of 64.7 cm , when dropped from a height of 100 cm . A change is made in the manufacturing process and it is required to test whether this change has affected the mean height of bounce. 100 new tennis balls are tested and it is found that their mean height of bounce when dropped from a height of 100 cm is 65.7 cm and the unbiased estimate of the population variance is \(15 \mathrm {~cm} ^ { 2 }\).
    1. Calculate a \(95 \%\) confidence interval for the population mean.
    2. Use your answer to part (ii) (a) to explain what conclusion can be drawn about whether the change has affected the mean height of bounce.
CAIE S2 2016 June Q4
6 marks Standard +0.3
4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the \(5 \%\) significance level whether the mean rate has changed.
CAIE S2 2016 June Q5
10 marks Standard +0.8
5 Each box of Fruity Flakes contains \(X\) grams of flakes and \(Y\) grams of fruit, where \(X\) and \(Y\) are independent random variables, having distributions \(\mathrm { N } ( 400,50 )\) and \(\mathrm { N } ( 100,20 )\) respectively. The weight of each box, when empty, is exactly 20 grams. A full box of Fruity Flakes is chosen at random.
  1. Find the probability that the total weight of the box and its contents is less than 530 grams.
  2. Find the probability that the weight of flakes in the box is more than 4.1 times the weight of fruit in the box.
CAIE S2 2016 June Q6
10 marks Moderate -0.3
6 At a certain shop the demand for hair dryers has a Poisson distribution with mean 3.4 per week.
  1. Find the probability that, in a randomly chosen two-week period, the demand is for exactly 5 hair dryers.
  2. At the beginning of a week the shop has a certain number of hair dryers for sale. Find the probability that the shop has enough hair dryers to satisfy the demand for the week if
    1. they have 4 hair dryers in the shop,
    2. they have 5 hair dryers in the shop.
    3. Find the smallest number of hair dryers that the shop needs to have at the beginning of a week so that the probability of being able to satisfy the demand that week is at least 0.9 .
CAIE S2 2016 June Q7
11 marks Moderate -0.3
7
  1. \includegraphics[max width=\textwidth, alt={}, center]{1060d9f5-cf40-419e-b212-7266885c6617-3_465_1127_954_550} The diagram shows the graph of the probability density function of a variable \(X\). Given that the graph is symmetrical about the line \(x = 1\) and that \(\mathrm { P } ( 0 < X < 2 ) = 0.6\), find \(\mathrm { P } ( X > 0 )\).
  2. A flower seller wishes to model the length of time that tulips last when placed in a jug of water. She proposes a model using the random variable \(X\) (in hundreds of hours) with probability density function given by $$f ( x ) = \begin{cases} k \left( 2.25 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1.5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
    1. Show that \(k = \frac { 4 } { 9 }\).
    2. Use this model to find the mean number of hours that a tulip lasts in a jug of water. The flower seller wishes to create a similar model for daffodils. She places a large number of daffodils in jugs of water and the longest time that any daffodil lasts is found to be 290 hours.
    3. Give a reason why \(\mathrm { f } ( x )\) would not be a suitable model for daffodils.
    4. The flower seller considers a model for daffodils of the form $$g ( x ) = \begin{cases} c \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(c\) are constants. State a suitable value for \(a\). (There is no need to evaluate \(c\).)
CAIE S2 2017 June Q1
5 marks Moderate -0.8
1 On average, 1 clover plant in 10000 has four leaves instead of three.
  1. Use an approximating distribution to calculate the probability that, in a random sample of 2000 clover plants, more than 2 will have four leaves.
  2. Justify your approximating distribution.
CAIE S2 2017 June Q2
6 marks Moderate -0.3
2 Past experience has shown that the heights of a certain variety of plant have mean 64.0 cm and standard deviation 3.8 cm . During a particularly hot summer, it was expected that the heights of plants of this variety would be less than usual. In order to test whether this was the case, a botanist recorded the heights of a random sample of 100 plants and found that the value of the sample mean was 63.3 cm . Stating a necessary assumption, carry out the test at the \(2.5 \%\) significance level.
CAIE S2 2017 June Q3
6 marks Standard +0.8
3
  1. The waiting time at a certain bus stop has variance 2.6 minutes \({ } ^ { 2 }\). For a random sample of 75 people, the mean waiting time was 7.1 minutes. Calculate a \(92 \%\) confidence interval for the population mean waiting time.
  2. A researcher used 3 random samples to calculate 3 independent \(92 \%\) confidence intervals. Find the probability that all 3 of these confidence intervals contain only values that are greater than the actual population mean.
  3. Another researcher surveyed the first 75 people who waited at a bus stop on a Monday morning. Give a reason why this sample is unsuitable for use in finding a confidence interval for the mean waiting time.
CAIE S2 2017 June Q4
9 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{332f0909-c192-40f7-88b7-7cfec2db2eef-06_428_773_260_685} The time, \(X\) minutes, taken by a large number of runners to complete a certain race has probability density function f given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant, as shown in the diagram.
  1. Without calculation, explain how you can tell that there were more runners whose times were below 7.5 minutes than above 7.5 minutes.
  2. Show that \(k = 10\).
  3. Find \(\mathrm { E } ( X )\).
  4. Find \(\operatorname { Var } ( X )\).
CAIE S2 2017 June Q5
10 marks Standard +0.3
5 Large packets of sugar are packed in cartons, each containing 12 packets. The weights of these packets are normally distributed with mean 505 g and standard deviation 3.2 g . The weights of the cartons, when empty, are independently normally distributed with mean 150 g and standard deviation 7 g .
  1. Find the probability that the total weight of a full carton is less than 6200 g .
    Small packets of sugar are packed in boxes. The total weight of a full box has a normal distribution with mean 3130 g and standard deviation 12.1 g .
  2. Find the probability that the weight of a randomly chosen full carton is less than double the weight of a randomly chosen full box.
CAIE S2 2017 June Q6
14 marks Standard +0.3
6 The number of sports injuries per month at a certain college has a Poisson distribution. In the past the mean has been 1.1 injuries per month. The principal recently introduced new safety guidelines and she decides to test, at the \(2 \%\) significance level, whether the mean number of sports injuries has been reduced. She notes the number of sports injuries during a 6-month period.
  1. Find the critical region for the test and state the probability of a Type I error.
  2. State what is meant by a Type I error in this context.
  3. During the 6 -month period there are a total of 2 sports injuries. Carry out the test.
  4. Assuming that the mean remains 1.1 , calculate the probability that there will be fewer than 30 sports injuries during a 36-month period.
CAIE S2 2017 June Q1
4 marks Easy -1.2
1 In a survey of 2000 randomly chosen adults, 1602 said that they owned a smartphone. Calculate an approximate \(95 \%\) confidence interval for the proportion of adults in the whole population who own a smartphone.
CAIE S2 2017 June Q2
5 marks Moderate -0.3
2 Javier writes an article containing 52460 words. He plans to upload the article to his website, but he knows that this process sometimes introduces errors. He assumes that for each word in the uploaded version of his article, the probability that it contains an error is 0.00008 . The number of words containing an error is denoted by \(X\).
  1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\), giving your answers correct to three decimal places.
    Javier wants to use the Poisson distribution as an approximating distribution to calculate the probability that there will be fewer than 5 words containing an error in his uploaded article.
  2. Explain how your answers to part (i) are consistent with the use of the Poisson distribution as an approximating distribution.
  3. Use the Poisson distribution to calculate \(\mathrm { P } ( X < 5 )\).
CAIE S2 2017 June Q3
8 marks Standard +0.3
3 Household incomes, in thousands of dollars, in a certain country are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The incomes of a random sample of 400 households are found and the results are summarised below. $$n = 400 \quad \Sigma x = 923 \quad \Sigma x ^ { 2 } = 3170$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. A random sample of 50 households in one particular region of the country is taken and the sample mean income, in thousands of dollars, is found to be 2.6 . Using your values from part (i), test at the \(5 \%\) significance level whether household incomes in this region are greater, on average, than in the country as a whole.
CAIE S2 2017 June Q4
7 marks Moderate -0.3
4 It is claimed that 1 in every 4 packets of certain biscuits contains a free gift. Marisa and André both suspect that the true proportion is less than 1 in 4.
  1. Marisa chooses 20 packets at random. She decides that if fewer than 3 contain free gifts, she will conclude that the claim is not justified. Use a binomial distribution to find the probability of a Type I error.
  2. André chooses 25 packets at random. He decides to carry out a significance test at the \(1 \%\) level, using a binomial distribution. Given that only 1 of the 25 packets contains a free gift, carry out the test.
CAIE S2 2017 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{c06524f0-a981-48a6-9af0-c4a3474396b3-06_394_723_258_705} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and \(a\) only. It is given that \(\mathrm { P } ( X < 1 ) = 0.25\).
  1. Find, in any order,
    1. \(\mathrm { P } ( X < 2 )\),
    2. the value of \(a\),
    3. \(\mathrm { f } ( x )\).
    4. Find the median of \(X\).
CAIE S2 2017 June Q6
9 marks Standard +0.3
6 Old televisions arrive randomly and independently at a recycling centre at an average rate of 1.2 per day.
  1. Find the probability that exactly 2 televisions arrive in a 2-day period.
  2. Use an appropriate approximating distribution to find the probability that at least 55 televisions arrive in a 50-day period.
    Independently of televisions, old computers arrive randomly and independently at the same recycling centre at an average rate of 4 per 7-day week.
  3. Find the probability that the total number of televisions and computers that arrive at the recycling centre in a 3-day period is less than 4.