Questions — OCR S2 (167 questions)

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OCR S2 2015 June Q4
4 A continuous random variable is normally distributed with mean \(\mu\). A significance test for \(\mu\) is carried out, at the \(5 \%\) significance level, on 90 independent occasions.
  1. Given that the null hypothesis is correct on all 90 occasions, use a suitable approximation to find the probability that on 6 or fewer occasions the test results in a Type I error. Justify your approximation.
  2. Given instead that on all 90 occasions the probability of a Type II error is 0.35 , use a suitable approximation to find the probability that on fewer than 29 occasions the test results in a Type II error.
OCR S2 2015 June Q5
5
  1. State an advantage of using random numbers in selecting samples.
  2. It is known that in analysing the digits in large sets of financial records, the probability that the leading digit is 1 is 0.25 . A random sample of 18 leading digits from a certain large set of financial records is obtained and it is found that 8 of the leading digits are 1 s . Test, at the \(5 \%\) significance level, whether the probability that the leading digit is 1 in this set of records is greater than 0.25 .
OCR S2 2015 June Q6
6 Records for a doctors' surgery over a long period suggest that the time taken for a consultation, \(T\) minutes, has a mean of 11.0. Following the introduction of new regulations, a doctor believes that the average time has changed. She finds that, with new regulations, the consultation times for a random sample of 120 patients can be summarised as $$n = 120 , \Sigma t = 1411.20 , \Sigma t ^ { 2 } = 18737.712 .$$
  1. Test, at the \(10 \%\) significance level, whether the doctor's belief is correct.
  2. Explain whether, in answering part (i), it was necessary to assume that the consultation times were normally distributed.
OCR S2 2015 June Q7
7 A large railway network suffers points failures at an average rate of 1 every 3 days. Assume that the number of points failures can be modelled by a Poisson distribution. The network employs a new firm of engineers. After the new engineers have become established, it is found that in a randomly chosen period of 15 days there are 2 instances of points failures.
  1. Test, at the \(5 \%\) significance level, whether there is evidence that the mean number of points failures has been reduced.
  2. A new test is carried out over a period of 150 days. Use a suitable approximation to find the greatest number of points failures there could be in 150 days that would lead to a \(5 \%\) significance test concluding that the average number of points failures had been reduced.
OCR S2 2015 June Q8
8 The random variable \(S\) has the distribution \(\mathrm { B } ( 14 , p )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.3\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p > 0.3\). The critical region for the test is \(S \geqslant 8\).
  1. Find the significance level of the test, correct to 3 significant figures.
  2. It is given that, on each occasion that the test is carried out, the true value of \(p\) is equally likely to be \(0.3,0.5\) or 0.7 , independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.
OCR S2 2016 June Q1
1 The results of 14 observations of a random variable \(V\) are summarised by $$n = 14 , \quad \sum v = 3752 , \quad \sum v ^ { 2 } = 1007448 .$$ Calculate unbiased estimates of \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
OCR S2 2016 June Q2
2 The mass, in kilograms, of a packet of flour is a normally distributed random variable with mean 1.03 and variance \(\sigma ^ { 2 }\). Given that \(5 \%\) of packets have mass less than 1.00 kg , find the percentage of packets with mass greater than 1.05 kg .
OCR S2 2016 June Q3
3 The random variable \(F\) has the distribution \(\mathrm { B } ( 40,0.65 )\). Use a suitable approximation to find \(\mathrm { P } ( F \leqslant 30 )\), justifying your approximation.
OCR S2 2016 June Q4
4 It is given that \(Y \sim \operatorname { Po } ( \lambda )\), where \(\lambda \neq 0\), and that \(\mathrm { P } ( Y = 4 ) = \mathrm { P } ( Y = 5 )\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of \(\mathrm { P } ( Y = 5 )\).
\(555 \%\) of the pupils in a large school are girls. A member of the student council claims that the probability that a girl rather than a boy becomes Head Student is greater than 0.55 . As evidence for his claim he says that 6 of the last 8 Head Students have been girls.
  1. Use an exact binomial distribution to test the claim at the \(10 \%\) significance level.
  2. A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid.
OCR S2 2016 June Q6
6 The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).
  1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution.
  2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate \(\mathrm { P } ( 4 \leqslant X < 8 )\).
  3. At a busy time of the day, \(\lambda = 30\).
    (a) Assuming that a Poisson distribution is valid, use a suitable approximation to find \(\mathrm { P } ( X > 35 )\). Justify your approximation.
    (b) Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\).
OCR S2 2016 June Q7
7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } a x ^ { - 3 } + b x ^ { - 4 } & x \geqslant 1
0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.
  1. Explain what the letter \(x\) represents. It is given that \(\mathrm { P } ( X > 2 ) = \frac { 3 } { 16 }\).
  2. Show that \(a = 1\), and find the value of \(b\).
  3. Find \(\mathrm { E } ( X )\).
OCR S2 2016 June Q8
8 It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years \(^ { 2 }\).
  1. Test at the \(5 \%\) significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild.
  2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years \({ } ^ { 2 }\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test.
  3. Explain whether the Central Limit Theorem is needed in these tests.
OCR S2 2016 June Q9
9 The random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\). A significance test is carried out at the \(1 \%\) level of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 11\) against \(\mathrm { H } _ { 1 } : \lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14 , find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. \section*{END OF QUESTION PAPER}
OCR S2 2009 January Q1
1 A newspaper article consists of 800 words. For each word, the probability that it is misprinted is 0.005 , independently of all other words. Use a suitable approximation to find the probability that the total number of misprinted words in the article is no more than 6 . Give a reason to justify your approximation.
OCR S2 2009 January Q2
2 The continuous random variable \(Y\) has the distribution \(\mathrm { N } \left( 23.0,5.0 ^ { 2 } \right)\). The mean of \(n\) observations of \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } > 23.625 ) = 0.0228\). Find the value of \(n\).
OCR S2 2009 January Q3
3 The number of incidents of radio interference per hour experienced by a certain listener is modelled by a random variable with distribution \(\operatorname { Po } ( 0.42 )\).
  1. Find the probability that the number of incidents of interference in one randomly chosen hour is
    (a) 0 ,
    (b) exactly 1 .
  2. Find the probability that the number of incidents in a randomly chosen 5-hour period is greater than 3.
  3. One hundred hours of listening are monitored and the numbers of 1 -hour periods in which 0,1 , \(2 , \ldots\) incidents of interference are experienced are noted. A bar chart is drawn to represent the results. Without any further calculations, sketch the shape that you would expect for the bar chart. (There is no need to use an exact numerical scale on the frequency axis.)
OCR S2 2009 January Q4
4 A television company believes that the proportion of adults who watched a certain programme is 0.14 . Out of a random sample of 22 adults, it is found that 2 watched the programme.
  1. Carry out a significance test, at the \(10 \%\) level, to determine, on the basis of this sample, whether the television company is overestimating the proportion of adults who watched the programme.
  2. The sample was selected randomly. State what properties of this method of sampling are needed to justify the use of the distribution used in your test.
OCR S2 2009 January Q5
5 The continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}
T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases} \end{array}$$
  1. Sketch, on the same axes, the graphs of f and g .
  2. Describe in everyday terms the difference between the distributions of the random variables \(S\) and \(T\). (Answers that comment only on the shapes of the graphs will receive no credit.)
  3. Calculate the variance of \(T\).
OCR S2 2009 January Q6
6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).
  1. Given that \(n = 81\),
    (a) find the critical region for the test, in terms of the sample mean \(\bar { W }\),
    (b) find the probability that the test results in a Type II error when \(\mu = 50.2\).
  2. State how the probability of this Type II error would change if \(n\) were greater than 81 .
OCR S2 2009 January Q7
7 A motorist records the time taken, \(T\) minutes, to drive a particular stretch of road on each of 64 occasions. Her results are summarised by $$\Sigma t = 876.8 , \quad \Sigma t ^ { 2 } = 12657.28$$
  1. Test, at the \(5 \%\) significance level, whether the mean time for the motorist to drive the stretch of road is greater than 13.1 minutes.
  2. Explain whether it is necessary to use the Central Limit Theorem in your test.
OCR S2 2009 January Q8
8 A sales office employs 21 representatives. Each day, for each representative, the probability that he or she achieves a sale is 0.7 , independently of other representatives. The total number of representatives who achieve a sale on any one day is denoted by \(K\).
  1. Using a suitable approximation (which should be justified), find \(\mathrm { P } ( K \geqslant 16 )\).
  2. Using a suitable approximation (which should be justified), find the probability that the mean of 36 observations of \(K\) is less than or equal to 14.0 . 4
OCR S2 2011 January Q1
1 A random sample of nine observations of a random variable is obtained. The results are summarised as $$\Sigma x = 468 , \quad \Sigma x ^ { 2 } = 24820 .$$ Calculate unbiased estimates of the population mean and variance.
OCR S2 2011 January Q2
2 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). The mean of a sample of \(n\) observations of \(H\) is denoted by \(\bar { H }\). It is given that \(\mathrm { P } ( \bar { H } > 53.28 ) = 0.0250\) and \(\mathrm { P } ( \bar { H } < 51.65 ) = 0.0968\), both correct to 4 decimal places. Find the values of \(\mu\) and \(n\).
OCR S2 2011 January Q3
3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.
OCR S2 2011 January Q4
4 The continuous random variable \(X\) has mean \(\mu\) and standard deviation 45. A significance test is to be carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 230\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 230\), at the \(1 \%\) significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.
  1. Carry out the test.
  2. Explain whether it is necessary to use the Central Limit Theorem in your test.