Questions S2 (1597 questions)

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CAIE S2 2016 November Q2
5 marks Standard +0.3
2 A die has six faces numbered \(1,2,3,4,5,6\). Manjit suspects that the die is biased so that it shows a six on fewer throws than it would if it were fair. In order to test her suspicion, she throws the die a certain number of times and counts the number of sixes.
  1. State suitable null and alternative hypotheses for Manjit's test.
  2. There are no sixes in the first 15 throws. Show that this result is not significant at the \(5 \%\) level.
  3. Find the smallest value of \(n\) such that, if there are no sixes in the first \(n\) throws, this result is significant at the 5\% level.
CAIE S2 2016 November Q3
7 marks Standard +0.3
3 Particles are emitted randomly from a radioactive substance at a constant average rate of 3.6 per minute. Find the probability that
  1. more than 3 particles are emitted during a 20 -second period,
  2. more than 240 particles are emitted during a 1-hour period.
CAIE S2 2016 November Q4
7 marks Standard +0.3
4 Each week a farmer sells \(X\) litres of milk and \(Y \mathrm {~kg}\) of cheese, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 1520,53 ^ { 2 } \right)\) and \(\mathrm { N } \left( 175,12 ^ { 2 } \right)\) respectively.
  1. Find the mean and standard deviation of the total amount of milk that the farmer sells in 4 randomly chosen weeks. During a year when milk prices are low, the farmer makes a loss of 2 cents per litre on milk and makes a profit of 21 cents per kg on cheese, so the farmer's overall weekly profit is \(( 21 Y - 2 X )\) cents.
  2. Find the probability that, in a randomly chosen week, the farmer's overall profit is positive.
CAIE S2 2016 November Q5
8 marks Standard +0.3
5
  1. The masses, in grams, of certain tomatoes are normally distributed with standard deviation 9 grams. A random sample of 100 tomatoes has a sample mean of 63 grams. Find a \(90 \%\) confidence interval for the population mean mass of these tomatoes.
  2. The masses, in grams, of certain potatoes are normally distributed with known population standard deviation but unknown population mean. A random sample of potatoes is taken in order to find a confidence interval for the population mean. Using a sample of size 50 , a \(95 \%\) confidence interval is found to have width 8 grams.
    1. Using another sample of size 50 , an \(\alpha \%\) confidence interval has width 4 grams. Find \(\alpha\).
    2. Find the sample size \(n\), such that a \(95 \%\) confidence interval has width 4 grams.
CAIE S2 2016 November Q6
9 marks Moderate -0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_504_260_534}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_373_495_260_1123}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_497_776_534}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ (a) Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    (b) Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    (c) Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q7
11 marks Standard +0.3
7 In the past the time, in minutes, taken for a particular rail journey has been found to have mean 20.5 and standard deviation 1.2. Some new railway signals are installed. In order to test whether the mean time has decreased, a random sample of 100 times for this journey are noted. The sample mean is found to be 20.3 minutes. You should assume that the standard deviation is unchanged.
  1. Carry out a significance test, at the \(4 \%\) level, of whether the population mean time has decreased. Later another significance test of the same hypotheses, using another random sample of size 100 , is carried out at the \(4 \%\) level.
  2. Given that the population mean is now 20.1, find the probability of a Type II error.
  3. State what is meant by a Type II error in this context.
CAIE S2 2016 November Q6
9 marks Moderate -0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_504_260_534}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_373_495_260_1123}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_497_776_534}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ (a) Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    (b) Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    (c) Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q1
3 marks Easy -1.2
1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).
CAIE S2 2016 November Q2
3 marks Easy -1.8
2 Dominic wishes to choose a random sample of five students from the 150 students in his year. He numbers the students from 1 to 150 . Then he uses his calculator to generate five random numbers between 0 and 1 . He multiplies each random number by 150 and rounds up to the next whole number to give a student number.
  1. Dominic's first random number is 0.392 . Find the student number that is produced by this random number.
  2. Dominic's second student number is 104 . Find a possible random number that would produce this student number.
  3. Explain briefly why five random numbers may not be enough to produce a sample of five student numbers.
CAIE S2 2016 November Q3
5 marks Standard +0.3
3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.
CAIE S2 2016 November Q4
5 marks Challenging +1.2
4 The manufacturer of a tablet computer claims that the mean battery life is 11 hours. A consumer organisation wished to test whether the mean is actually greater than 11 hours. They invited a random sample of members to report the battery life of their tablets. They then calculated the sample mean. Unfortunately a fire destroyed the records of this test except for the following partial document.
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-2_467_593_1612_776} Given that the population of battery lives is normally distributed with standard deviation 1.6 hours, find the set of possible values of the sample size, \(n\).
CAIE S2 2016 November Q5
8 marks Moderate -0.3
5 It is claimed that \(30 \%\) of packets of Froogum contain a free gift. Andre thinks that the actual proportion is less than \(30 \%\) and he decides to carry out a hypothesis test at the \(5 \%\) significance level. He buys 20 packets of Froogum and notes the number of free gifts he obtains.
  1. State null and alternative hypotheses for the test.
  2. Use a binomial distribution to find the probability of a Type I error. Andre finds that 3 of the 20 packets contain free gifts.
  3. Carry out the test.
CAIE S2 2016 November Q6
8 marks Standard +0.3
6 A variable \(X\) takes values \(1,2,3,4,5\), and these values are generated at random by a machine. Each value is supposed to be equally likely, but it is suspected that the machine is not working properly. A random sample of 100 values of \(X\), generated by the machine, gives the following results. $$n = 100 \quad \Sigma x = 340 \quad \Sigma x ^ { 2 } = 1356$$
  1. Find a 95\% confidence interval for the population mean of the values generated by the machine.
  2. Use your answer to part (i) to comment on whether the machine may be working properly.
CAIE S2 2016 November Q7
9 marks Standard +0.3
7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.
  1. Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.
  2. Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.
CAIE S2 2016 November Q8
9 marks Standard +0.3
8
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\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between - 3 and 3 only, and their standard deviations are \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) respectively.
  1. List \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ (a) Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
    (b) Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
    (c) Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).
OCR S2 2007 January Q1
4 marks Moderate -0.8
1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).
OCR S2 2007 January Q2
5 marks Moderate -0.8
2 A school has 900 pupils. For a survey, Jan obtains a list of all the pupils, numbered 1 to 900 in alphabetical order. She then selects a sample by the following method. Two fair dice, one red and one green, are thrown, and the number in the list of the first pupil in the sample is determined by the following table.
\cline { 3 - 8 } \multicolumn{2}{c|}{}Score on green dice
\cline { 3 - 8 } \multicolumn{2}{c|}{}123456
Score on
red dice
1,2 or 3123456
For example, if the scores on the red and green dice are 5 and 2 respectively, then the first member of the sample is the pupil numbered 8 in the list. Starting with this first number, every 12th number on the list is then used, so that if the first pupil selected is numbered 8 , the others will be numbered \(20,32,44 , \ldots\).
  1. State the size of the sample.
  2. Explain briefly whether the following statements are true.
    (a) Each pupil in the school has an equal probability of being in the sample.
    (b) The pupils in the sample are selected independently of one another.
  3. Give a reason why the number of the first pupil in the sample should not be obtained simply by adding together the scores on the two dice. Justify your answer.
OCR S2 2007 January Q3
6 marks Moderate -0.5
3 A fair dice is thrown 90 times. Use an appropriate approximation to find the probability that the number 1 is obtained 14 or more times.
OCR S2 2007 January Q4
7 marks Moderate -0.8
4 A set of observations of a random variable \(W\) can be summarised as follows: $$n = 14 , \quad \Sigma w = 100.8 , \quad \Sigma w ^ { 2 } = 938.70 .$$
  1. Calculate an unbiased estimate of the variance of \(W\).
  2. The mean of 70 observations of \(W\) is denoted by \(\bar { W }\). State the approximate distribution of \(\bar { W }\), including unbiased estimate(s) of any parameter(s).
OCR S2 2007 January Q5
12 marks Standard +0.3
5 On a particular night, the number of shooting stars seen per minute can be modelled by the distribution \(\operatorname { Po(0.2). }\)
  1. Find the probability that, in a given 6 -minute period, fewer than 2 shooting stars are seen.
  2. Find the probability that, in 20 periods of 6 minutes each, the number of periods in which fewer than 2 shooting stars are seen is exactly 13 .
  3. Use a suitable approximation to find the probability that, in a given 2-hour period, fewer than 30 shooting stars are seen.
OCR S2 2007 January Q6
13 marks Standard +0.3
6 The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(2 a + 2 b = 1\).
  2. It is given that \(\mathrm { E } ( X ) = \frac { 11 } { 9 }\). Use this information to find a second equation connecting \(a\) and \(b\), and hence find the values of \(a\) and \(b\).
  3. Determine whether the median of \(X\) is greater than, less than, or equal to \(\mathrm { E } ( X )\).
OCR S2 2007 January Q7
11 marks Standard +0.3
7 A television company believes that the proportion of households that can receive Channel C is 0.35 .
  1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35.
  2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
OCR S2 2007 January Q8
14 marks Challenging +1.8
8 The quantity, \(X\) milligrams per litre, of silicon dioxide in a certain brand of mineral water is a random variable with distribution \(\mathrm { N } \left( \mu , 5.6 ^ { 2 } \right)\).
  1. A random sample of 80 observations of \(X\) has sample mean 100.7. Test, at the \(1 \%\) significance level, the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 102\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 102\).
  2. The test is redesigned so as to meet the following conditions.
    • The hypotheses are \(\mathrm { H } _ { 0 } : \mu = 102\) and \(\mathrm { H } _ { 1 } : \mu < 102\).
    • The significance level is \(1 \%\).
    • The probability of making a Type II error when \(\mu = 100\) is to be (approximately) 0.05 .
    The sample size is \(n\), and the critical region is \(\bar { X } < c\), where \(\bar { X }\) denotes the sample mean.
    (a) Show that \(n\) and \(c\) satisfy (approximately) the equation \(102 - c = \frac { 13.0256 } { \sqrt { n } }\).
    (b) Find another equation satisfied by \(n\) and \(c\).
    (c) Hence find the values of \(n\) and \(c\).
OCR S2 2008 January Q1
6 marks Standard +0.3
1 The random variable \(T\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( T > 80 ) = 0.05\) and \(\mathrm { P } ( T > 50 ) = 0.75\). Find the values of \(\mu\) and \(\sigma\).
OCR S2 2008 January Q2
5 marks Moderate -0.8
2 A village has a population of 600 people. A sample of 12 people is obtained as follows. A list of all 600 people is obtained and a three-digit number, between 001 and 600 inclusive, is allocated to each name in alphabetical order. Twelve three-digit random numbers, between 001 and 600 inclusive, are obtained and the people whose names correspond to those numbers are chosen.
  1. Find the probability that all 12 of the numbers chosen are 500 or less.
  2. When the selection has been made, it is found that all of the numbers chosen are 500 or less. One of the people in the village says, "The sampling method must have been biased." Comment on this statement.