Linear combinations of normal random variables

332 questions · 38 question types identified

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Two or more different variables

Questions asking for the probability of a linear combination involving two or more different normal variables (e.g. aX + bY < k where X and Y have different distributions).

25 Standard +0.4
7.5% of questions
5.04b Linear combinations: of normal distributions
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2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 6.5,14 )\) and \(\mathrm { N } ( 7.4,15 )\) respectively. Find \(\mathrm { P } ( 3 X - Y < 20 )\).
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Easiest question Moderate -0.5 »
2. A workshop makes two types of electrical resistor. The resistance, \(X\) ohms, of resistors of Type A is such that \(X \sim \mathrm {~N} ( 20,4 )\).
The resistance, \(Y\) ohms, of resistors of Type B is such that \(Y \sim \mathrm {~N} ( 10,0.84 )\).
When a resistor of each type is connected into a circuit, the resistance \(R\) ohms of the circuit is given by \(R = X + Y\) where \(X\) and \(Y\) are independent. Find
  1. \(\mathrm { E } ( R )\),
  2. \(\operatorname { Var } ( R )\),
  3. \(\mathrm { P } ( 28.9 < R < 32.64 )\) (6)
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Hardest question Challenging +1.2 »
  1. The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.
Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)
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Comparison involving sums or multiples

Questions asking for comparisons where at least one side involves a sum of multiple variables (e.g., P(aX₁ + bX₂ > cY₁ + dY₂)) or comparing totals of several items, requiring linear combinations of more than two base variables.

21 Standard +0.4
6.3% of questions
5.04b Linear combinations: of normal distributions
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3 The lengths, in centimetres, of two types of insect, \(A\) and \(B\), are modelled by the random variables \(X \sim \mathrm {~N} ( 6.2,0.36 )\) and \(Y \sim \mathrm {~N} ( 2.4,0.25 )\) respectively. Find the probability that the length of a randomly chosen type \(A\) insect is greater than the sum of the lengths of 3 randomly chosen type \(B\) insects.
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Easiest question Moderate -0.3 »
  1. The random variable \(A\) is defined as
$$A = 4 X - 3 Y$$ where \(X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent. Find
  1. \(\mathrm { E } ( A )\),
  2. \(\operatorname { Var } ( A )\). The random variables \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\) and \(Y _ { 4 }\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
  3. Find \(\mathrm { P } ( B > A )\).
    advancing learning, changing lives
    1. A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.
    Test, at the \(5 \%\) level of significance, the managing director's claim. State your hypotheses clearly.
    2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.
  4. Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
  5. find the probability that Philip beats James in the race by more than 2 minutes.
    3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .
  6. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
  7. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
  8. find a \(98 \%\) confidence interval for \(w\).
    1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.
    advancing learning, changing lives \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-055_2632_1828_123_121}
    2. A county councillor is investigating the level of hardship, h , of a town and the number of calls per 100 people to the emergency services, c. He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
    1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
    \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-081_2642_1833_118_118}
    2. A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than \(5 \%\).
    (5)
    3. The table below shows the population and the number of council employees for different towns and villages. \end{table} A nswers without working may not gain full credit. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{ \(0 - 3\) & 8
    \hline \(3 - 5\) & 12
    \hline \(5 - 6\) & 13
    \hline \(6 - 8\) & 9
    \hline \(8 - 12\) & 8
    \hline \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  9. Show that an estimate of \(\bar { X } = 5.49\) and an estimate of \(S _ { X } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
    Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]
    Waiting Time\(\mathrm { x } < 3\)\(3 - 5\)\(5 - 6\)\(6 - 8\)\(\mathrm { x } > 8\)
    Expected Frequency8.5612.737.56ab
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  10. Find the value of a and the value of b .
  11. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.
    \section*{Q uestion 4 continued}
    1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
    One large and 3 small bottles of Blumen are chosen at random.
  12. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
  13. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
    \section*{Q uestion 5 continued} 6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
  14. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
  15. Explain the relevance of the Central Limit Theorem to the test in part (a). \section*{Q uestion 6 continued} \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
    7. Lambs are born in a shed on M ill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below. $$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
  16. Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
  17. Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
  18. Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
    \section*{Q uestion 7 continued} \end{figure}
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Hardest question Challenging +1.2 »
7. Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.
  1. Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.
  2. Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar. A random sample of 5 medium bags of sugar is taken.
  3. Find the value of \(d\) so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than \(d\) grams.
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Two-sample t-test (unknown variances)

Questions requiring a hypothesis test comparing two population means where population variances are unknown and must be estimated from sample data, typically using a two-sample t-test or pooled variance approach.

21 Standard +0.4
6.3% of questions
5.05c Hypothesis test: normal distribution for population mean
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6 The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of 40 races run by Tania gave the following results. $$n = 40 \quad \Sigma t = 560 \quad \Sigma t ^ { 2 } = 7850$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
    The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(\mathrm { N } ( 14.2,0.3 )\).
  2. Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.
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Easiest question Moderate -0.3 »
  1. A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full-time staff and 4000 part-time staff.
    1. Describe how a stratified sample of 200 staff could be taken.
    2. Explain an advantage of using a stratified sample rather than a simple random sample.
    A random sample of 80 full-time staff and an independent random sample of 80 part-time staff were given a test of policy awareness. The results are summarised in the table below.
    Mean score \(( \bar { x } )\)
    Variance of
    scores \(\left( s ^ { 2 } \right)\)
    Full-time staff5221
    Part-time staff5019
  2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean policy awareness scores for full-time and part-time staff are different.
  3. Explain the significance of the Central Limit Theorem to the test in part (c).
  4. State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full-time staff and the 80 part-time staff were given another test of policy awareness. The value of the test statistic \(z\) was 2.53
  5. Comment on the awareness of company policy for the full-time and part-time staff in light of this result. Use a \(1 \%\) level of significance.
  6. Interpret your answers to part (c) and part (f).
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Hardest question Challenging +1.2 »
6. The carbon content, measured in suitable units, of steel is normally distributed. Two independent random samples of steel were taken from a refining plant at different times and their carbon content recorded. The results are given below. Sample A: \(\quad 1.5 \quad 0.9 \quad 1.3 \quad 1.2\) \(\begin{array} { l l l l l l l } \text { Sample } B : & 0.4 & 0.6 & 0.8 & 0.3 & 0.5 & 0.4 \end{array}\)
  1. Stating your hypotheses clearly, carry out a suitable test, at the \(10 \%\) level of significance, to show that both samples can be assumed to have come from populations with a common variance \(\sigma ^ { 2 }\).
  2. Showing your working clearly, find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on both samples.
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Multiple stage process probability

A question is this type if and only if it involves finding the probability for a process with 3+ independent stages where times/amounts are normally distributed (e.g. triathlon, multi-stage journey, game rounds).

17 Standard +0.6
5.1% of questions
5.04b Linear combinations: of normal distributions
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4 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions \(\mathrm { N } \left( 1020,45 ^ { 2 } \right)\) and \(\mathrm { N } \left( 2800,52 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total of the lifetimes of five randomly chosen Longlive bulbs is less than 5200 hours.
  2. Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least three times that of a randomly chosen Longlive bulb.
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Easiest question Moderate -0.3 »
2 In athletics matches the triple jump event consists of a hop, followed by a step, followed by a jump. The lengths covered by Albert in each part are independent normal variables with means \(3.5 \mathrm {~m} , 2.9 \mathrm {~m}\), 3.1 m and standard deviations \(0.3 \mathrm {~m} , 0.25 \mathrm {~m} , 0.35 \mathrm {~m}\) respectively. The length of the triple jump is the sum of the three parts.
  1. Find the mean and standard deviation of the length of Albert's triple jumps.
  2. Find the probability that the mean of Albert's next four triple jumps is greater than 9 m .
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Hardest question Challenging +1.2 »
6 When Sunil travels from his home in England to visit his relatives in India, his journey is in four stages. The times, in hours, for the stages have independent normal distributions as follows. Bus from home to the airport: \(\quad \mathrm { N } ( 3.75,1.45 )\) Waiting in the airport: \(\quad \mathrm { N } ( 3.1,0.785 )\) Flight from England to India: \(\quad \mathrm { N } ( 11,1.3 )\) Car in India to relatives: \(\quad \mathrm { N } ( 3.2,0.81 )\)
  1. Find the probability that the flight time is shorter than the total time for the other three stages.
  2. Find the probability that, for 6 journeys to India, the mean time waiting in the airport is less than 4 hours.
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Pure expectation and variance calculation

Questions that ask only to find E(aX + bY + c) and/or Var(aX + bY + c) with no further probability calculations or applications, where distributions are fully specified.

17 Moderate -0.2
5.1% of questions
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1 The independent random variables \(X\) and \(Y\) have standard deviations 3 and 6 respectively. Calculate the standard deviation of \(4 X - 5 Y\).
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Easiest question Easy -1.2 »
  1. \(\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c\),
  2. if \(X\) and \(Y\) are independent then \(\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )\).
\section*{Discrete distributions} \(X\) is a random variable taking values \(x _ { i }\) in a discrete distribution with \(\mathrm { P } \left( X = x _ { i } \right) = p _ { i }\) Expectation: \(\mu = \mathrm { E } ( X ) = \sum x _ { i } p _ { i }\) Variance: \(\sigma ^ { 2 } = \operatorname { Var } ( X ) = \sum \left( x _ { i } - \mu \right) ^ { 2 } p _ { i } = \sum x _ { i } ^ { 2 } p _ { i } - \mu ^ { 2 }\)
\(P ( X = x )\)E \(( X )\)\(\operatorname { Var } ( X )\)
Binomial \(\mathrm { B } ( n , p )\)\(\binom { n } { x } p ^ { x } ( 1 - p ) ^ { n - x }\)\(n p\)\(n p ( 1 - p )\)
Uniform distribution over \(1,2 , \ldots , n , \mathrm { U } ( n )\)\(\frac { 1 } { n }\)\(\frac { n + 1 } { 2 }\)\(\frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\)
Geometric distribution Geo(p)\(( 1 - p ) ^ { x - 1 } p\)\(\frac { 1 } { p }\)\(\frac { 1 - p } { p ^ { 2 } }\)
Poisson \(\operatorname { Po } ( \lambda )\)\(e ^ { - \lambda } \frac { \lambda ^ { x } } { x ! }\)\(\lambda\)\(\lambda\)
\section*{Continuous distributions} \(X\) is a continuous random variable with probability density function (p.d.f.) \(\mathrm { f } ( x )\) Expectation: \(\mu = \mathrm { E } ( X ) = \int x \mathrm { f } ( x ) \mathrm { d } x\) Variance: \(\sigma ^ { 2 } = \operatorname { Var } ( X ) = \int ( x - \mu ) ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \int x ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x - \mu ^ { 2 }\) Cumulative distribution function \(\mathrm { F } ( x ) = \mathrm { P } ( X \leq x ) = \int _ { - \infty } ^ { x } \mathrm { f } ( t ) \mathrm { d } t\)
p.d.f.E ( \(X\) )\(\operatorname { Var } ( X )\)
Continuous uniform distribution over [ \(a , b\) ]\(\frac { 1 } { b - a }\)\(\frac { 1 } { 2 } ( a + b )\)\(\frac { 1 } { 12 } ( b - a ) ^ { 2 }\)
Exponential\(\lambda \mathrm { e } ^ { - \lambda x }\)\(\frac { 1 } { \lambda }\)\(\frac { 1 } { \lambda ^ { 2 } }\)
Normal \(N \left( \mu , \sigma ^ { 2 } \right)\)\(\frac { 1 } { \sigma \sqrt { 2 \pi } } \mathrm { e } ^ { - \frac { 1 } { 2 } \left( \frac { x - \mu } { \sigma } \right) ^ { 2 } }\)\(\mu\)\(\sigma ^ { 2 }\)
\section*{Percentage points of the normal distribution} If \(Z\) has a normal distribution with mean 0 and variance 1 then, for each value of \(p\), the table gives the value of \(z\) such that \(P ( Z \leq z ) = p\).
\(p\)0.750.900.950.9750.990.9950.99750.9990.9995
\(z\)0.6741.2821.6451.9602.3262.5762.8073.0903.291
  1. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).
    a. Sketch the probability density function \(f ( x )\) of \(X\).
    b. State the value of \(\mathrm { P } ( X = 2 )\)
Find
c. \(\mathrm { E } ( X )\) d. \(\operatorname { Var } ( X )\)
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Hardest question Standard +0.3 »
1 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } \left( 10 , \sigma ^ { 2 } \right)\) and \(\operatorname { Po } ( 2 )\) respectively. The random variable \(S\) is given by \(S = 5 X - 2 Y + c\), where \(c\) is a constant.
It is given that \(\mathrm { E } ( S ) = \operatorname { Var } ( S ) = 408\).
  1. Find the value of \(c\) and show that \(\sigma ^ { 2 } = 16\).
  2. Find \(\mathrm { P } ( X \geqslant \mathrm { E } ( Y ) )\).
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Standard CI with summary statistics

Questions that provide sample sizes and either sample means/variances directly or summary statistics (Σx, Σx²) from which they must be calculated, using the standard normal approximation or t-distribution for the difference of means.

17 Standard +0.4
5.1% of questions
5.05d Confidence intervals: using normal distribution
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1 The number, \(x\), of pine trees was counted in each of 40 randomly chosen regions of equal size in country \(A\). The number, \(y\), of pine trees was counted in each of 60 randomly chosen regions of the same equal size in country \(B\). The results are summarised as follows. $$\sum x = 752 \quad \sum x ^ { 2 } = 14320 \quad \sum y = 1548 \quad \sum y ^ { 2 } = 40200$$ Find a 95\% confidence interval for the difference between the mean number of pine trees in regions of this size in countries \(A\) and \(B\).
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Easiest question Moderate -0.3 »
1 Kayla is investigating the lengths of the leaves of a certain type of tree found in two forests \(X\) and \(Y\). She chooses a random sample of 40 leaves of this type from forest \(X\) and records their lengths, \(x \mathrm {~cm}\). She also records the lengths, \(y \mathrm {~cm}\), for a random sample of 60 leaves of this type from forest \(Y\). Her results are summarised as follows. $$\sum x = 242.0 \quad \sum x ^ { 2 } = 1587.0 \quad \sum y = 373.2 \quad \sum y ^ { 2 } = 2532.6$$ Find a \(90 \%\) confidence interval for the difference between the population mean lengths of leaves in forests \(X\) and \(Y\).
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Hardest question Challenging +1.2 »
  1. Jamland and Goodjam are two suppliers of jars of jam. The weights of the jars of jam produced by each supplier can be assumed to be normally distributed with unknown, but equal, variances. A random sample of 20 jars of jam is taken from those supplied by Jamland.
Based on this sample, the 95\% confidence interval for the mean weight of a jar of Jamland jam, in grams, is
[0pt] [ 492, 507 ] A random sample of 10 jars of jam is selected from those supplied by Goodjam. The weight of each jar of Goodjam jam, \(y\) grams, is recorded. The results are summarised as follows $$\bar { y } = 480 \quad s _ { y } ^ { 2 } = 280$$ Find a 90\% confidence interval for the value by which the mean weight of a jar of jam supplied by Jamland exceeds the mean weight of a jar of jam supplied by Goodjam.
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Direct comparison with scalar multiple (different variables)

Questions asking for P(X > kY) or P(X < kY) where X and Y are from different distributions and k is a constant (including k=1 for different distributions), requiring distribution of X - kY.

16 Standard +0.8
4.8% of questions
5.04b Linear combinations: of normal distributions
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2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 31.2,10.4 ^ { 2 } \right)\). Two independent random values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen. Find \(\mathrm { P } \left( X _ { 1 } > 3 X _ { 2 } \right)\).
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Easiest question Standard +0.3 »
4 A random variable \(X\) has the distribution \(\mathrm { N } ( 10,12 )\). Two independent values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen at random.
  1. Write down the value of \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } \right)\).
  2. Find \(\mathrm { P } \left( X _ { 1 } > 2 X _ { 2 } - 3 \right)\).
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Hardest question Challenging +1.8 »
4 The heights of a certain variety of plant are normally distributed with mean 110 cm and variance \(1050 \mathrm {~cm} ^ { 2 }\). Two plants of this variety are chosen at random. Find the probability that the height of one of these plants is at least 1.5 times the height of the other.
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Single sum threshold probability

Questions finding the probability that the sum of observations from a single distribution exceeds or falls below a fixed threshold value (e.g., total weight of 20 bags exceeds 2 kg).

15 Standard +0.4
4.5% of questions
5.04b Linear combinations: of normal distributions
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3 The mass, in tonnes, of iron ore produced per day at a mine is normally distributed with mean 7.0 and standard deviation 0.46. Find the probability that the total amount of iron ore produced in 10 randomly chosen days is more than 71 tonnes.
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Easiest question Moderate -0.3 »
1 The masses, in grams, of apples of a certain type are normally distributed with mean 60.4 and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than 436 g is rejected. Find the proportion of bags that are rejected.
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Hardest question Challenging +1.2 »
7 The employees of a certain company have masses which are normally distributed. Female employees have a mean of 66.7 kg and standard deviation 9.3 kg , and male employees have a mean of 78.3 kg and standard deviation 8.5 kg . It may be assumed that all employees' masses are independent. On the ground floor 6 women and 9 men enter the empty staff lift for which it is stated that the maximum load is 1150 kg .
  1. Calculate the probability that the maximum load is exceeded. At the first floor all 15 passengers leave and 6 women, 8 men and an unknown employee enter.
  2. Assuming that the unknown employee is equally likely to be a woman or a man, calculate the probability that the maximum load is now exceeded.
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Expectation and variance with context application

Questions that find E and Var of linear combinations in a real-world context (costs, weights, measurements) where the linear combination represents a meaningful quantity like total cost or combined weight.

13 Moderate -0.1
3.9% of questions
5.04a Linear combinations: E(aX+bY), Var(aX+bY)
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2 An examination consists of a written paper and a practical test. The written paper marks ( \(M\) ) have mean 54.8 and standard deviation 16.0. The practical test marks ( \(P\) ) are independent of the written paper marks and have mean 82.4 and standard deviation 4.8. The final mark is found by adding \(75 \%\) of \(M\) to \(25 \%\) of \(P\). Find the mean and standard deviation of the final marks for the examination. [3]
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Easiest question Easy -1.2 »
1 At an internet café, the charge for using a computer is 5 cents per minute. The number of minutes for which people use a computer has mean 23 and standard deviation 8.
  1. Find, in cents, the mean and standard deviation of the amount people pay when using a computer.
  2. Each day, 15 people use computers independently. Find, in cents, the mean and standard deviation of the total amount paid by 15 people.
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Hardest question Standard +0.3 »
4 Each month a company sells \(X \mathrm {~kg}\) of brown sugar and \(Y \mathrm {~kg}\) of white sugar, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 2500,120 ^ { 2 } \right)\) and \(\mathrm { N } \left( 3700,130 ^ { 2 } \right)\) respectively.
  1. Find the mean and standard deviation of the total amount of sugar that the company sells in 3 randomly chosen months.
    The company makes a profit of \(\\) 1.50\( per kilogram of brown sugar sold and makes a loss of \)\\( 0.20\) per kilogram of white sugar sold.
  2. Find the probability that, in a randomly chosen month, the total profit is less than \(\\) 3000$.
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Mixed sum threshold probability

Questions finding the probability that the sum of observations from multiple different distributions combined exceeds or falls below a fixed threshold (e.g., 4 small bags plus 2 large bags totaling less than 4130g).

13 Standard +0.5
3.9% of questions
5.04b Linear combinations: of normal distributions
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3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution \(\mathrm { N } ( 250,10 )\). The mass, in grams, of one cup of flour has the independent distribution \(\mathrm { N } ( 160,9 )\). Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.
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Easiest question Standard +0.3 »
3 Mary buys 3 packets of sugar and 5 packets of coffee and puts them in her shopping basket, together with her purse which weighs 350 g . Weights of packets of sugar are normally distributed with mean 500 g and standard deviation 20 g . Weights of packets of coffee are normally distributed with mean 200 g and standard deviation 12 g . Find the probability that the total weight in the shopping basket is less than 2900 g .
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Hardest question Challenging +1.2 »
6 A particular lift has a maximum load capacity of 700 kg .
The weights of men are normally distributed with mean 80 kg and standard deviation 10 kg . The weights of women are normally distributed with mean 69 kg and standard deviation 5 kg . You may assume that weights of people are independent.
  1. Find the probability that when 6 men and 3 women are in the lift, the load exceeds 700 kg . A sign in the lift states: "Maximum number of people in the lift is \(c\) "
  2. Find the value of \(c\) such that the probability of the load exceeding 700 kg is less than \(2.5 \%\) no matter the gender of the occupants.
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Sum versus sum comparison

Questions comparing the total of m observations from one distribution against the total of n observations from a different distribution (e.g., 5 large bags vs 10 small bags).

12 Standard +0.7
3.6% of questions
5.04b Linear combinations: of normal distributions
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The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\text{N}(16.0, 0.4)\) and \(\text{N}(51.0, 0.9)\) respectively. Find the probability that the total mass of \(3\) randomly chosen small bags is greater than the mass of one randomly chosen large bag. [5]
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Easiest question Standard +0.3 »
6 The masses, in kilograms, of large and small sacks of grain have the distributions \(\mathrm { N } ( 53,11 )\) and \(\mathrm { N } ( 14,3 )\) respectively.
  1. Find the probability that the mass of a randomly chosen large sack is greater than four times the mass of a randomly chosen small sack.
  2. A lift can safely carry a maximum mass of 1000 kg . Find the probability that the lift can safely carry 12 randomly chosen large sacks and 25 randomly chosen small sacks. \(7 X\) is a random variable with distribution \(\operatorname { Po } ( 2.90 )\). A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is less than 2.88 .
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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Hardest question Challenging +1.2 »
7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm . The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm .
  1. Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. One large panel and one small panel are selected at random.
  2. Find the probability that the length of the large panel is more than \(\frac { 8 } { 3 }\) times the length of the small panel. Rosa needs 1000 cm of fencing. The large panels cost \(\pounds 80\) each and the small panels cost \(\pounds 30\) each. Rosa's plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.
  3. Calculate whether or not the expected cost of Rosa's plan is cheaper than simply buying 14 small panels.
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Pooled variance estimation

A question is this type if and only if it asks to find a pooled estimate of variance from two independent samples or to find a sample size given pooled variance information.

11 Standard +0.5
3.3% of questions
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6 The independent random variables \(X\) and \(Y\) have normal distributions with the same variance \(\sigma ^ { 2 }\). Samples of 5 observations of \(X\) and 10 observations of \(Y\) are made, and the results are summarised by \(\Sigma x = 15 , \Sigma x ^ { 2 } = 128 , \Sigma y = 36\) and \(\Sigma y ^ { 2 } = 980\). Find a pooled estimate of \(\sigma ^ { 2 }\).
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Easiest question Standard +0.3 »
8 The amounts spent on the weekly food shopping by families in the big city \(P\) and the small town \(Q\) are to be compared. The amounts spent, in dollars, in \(P\) and \(Q\) are denoted by \(x\) and \(y\) respectively. For a random sample of 60 families in \(P\) and a random sample of 50 families in \(Q\), the amounts are summarised as follows. $$\Sigma x = 9600 \quad \Sigma x ^ { 2 } = 1560000 \quad \Sigma y = 7200 \quad \Sigma y ^ { 2 } = 1052500$$ Assuming a common population variance, find
  1. a pooled estimate for the population variance,
  2. a \(95 \%\) confidence interval for the difference in the population means in \(P\) and \(Q\).
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Hardest question Challenging +1.2 »
6. A random sample \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 2 n }\) is taken from a population with mean \(\frac { \mu } { 3 }\) and variance \(3 \sigma ^ { 2 }\). A second random sample \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , \ldots , Y _ { n }\) is taken from a population with mean \(\frac { \mu } { 2 }\) and variance \(\frac { \sigma ^ { 2 } } { 2 }\), where the \(X\) and \(Y\) variables are all independent. \(A\), \(B\) and \(C\) are possible estimators of \(\mu\), where $$\begin{aligned} & A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 2 } \\ & B = \frac { 3 X _ { 1 } } { 2 } + \frac { 2 Y _ { 1 } } { 3 } \\ & C = \frac { 3 X _ { 1 } + 4 Y _ { 1 } } { 3 } \end{aligned}$$
  1. Show that two of \(A , B\) and \(C\) are unbiased estimators of \(\mu\) and find the bias of the third estimator of \(\mu\).
  2. Showing your working clearly, find which of \(A\), \(B\) and \(C\) is the best estimator of \(\mu\). The estimator $$D = \frac { 1 } { k } \left( \sum _ { i = 1 } ^ { 2 n } X _ { i } + \sum _ { i = 1 } ^ { n } Y _ { i } \right)$$ is an unbiased estimator of \(\mu\).
  3. Find \(k\) in terms of \(n\).
  4. Show that \(D\) is also a consistent estimator of \(\mu\).
  5. Find the least value of \(n\) for which \(D\) is a better estimator of \(\mu\) than any of \(A\), \(B\) or \(C\).
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Same variable, two observations

Questions where two independent observations are taken from the same normal distribution (same mean and standard deviation) and we find the probability their difference exceeds a threshold.

9 Standard +0.6
2.7% of questions
5.04b Linear combinations: of normal distributions
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4 The masses of a certain variety of potato are normally distributed with mean 180 g and variance \(1550 \mathrm {~g} ^ { 2 }\). Two potatoes of this variety are chosen at random. Find the probability that the mass of one of these potatoes is at least twice the mass of the other.
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Easiest question Standard +0.3 »
5
  1. Two random variables \(X\) and \(Y\) have the independent distributions \(\mathrm { N } ( 7,3 )\) and \(\mathrm { N } ( 6,2 )\) respectively. A random value of each variable is taken. Find the probability that the two values differ by more than 2 .
  2. Each candidate's overall score in a science test is calculated as follows. The mark for theory is denoted by \(T\), the mark for practical is denoted by \(P\), and the overall score is given by \(T + 1.5 P\). The variables \(T\) and \(P\) are assumed to be independent with distributions \(\mathrm { N } ( 62,158 )\) and \(\mathrm { N } ( 42,108 )\) respectively. You should assume that no continuity corrections are needed when using these distributions.
    1. A pass is awarded to candidates whose overall score is at least 90 . Find the proportion of candidates who pass.
    2. Comment on the assumption that the variables \(T\) and \(P\) are independent.
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Hardest question Challenging +1.8 »
4 The masses of a certain variety of potato are normally distributed with mean 180 g and variance \(1550 \mathrm {~g} ^ { 2 }\). Two potatoes of this variety are chosen at random. Find the probability that the mass of one of these potatoes is at least twice the mass of the other.
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Different variables, one observation each

Questions where one observation is taken from each of two different normal distributions (different means and/or standard deviations) and we find the probability their difference exceeds a threshold.

9 Standard +0.7
2.7% of questions
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4 A certain make of washing machine has a wash-time with mean 56.9 minutes and standard deviation 4.8 minutes. A certain make of tumble dryer has a drying-time with mean 61.1 minutes and standard deviation 6.3 minutes. Both times are normally distributed and are independent of each other. Find the probability that a randomly chosen wash-time differs by more than 3 minutes from a randomly chosen drying-time.
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Easiest question Standard +0.3 »
4 A certain make of washing machine has a wash-time with mean 56.9 minutes and standard deviation 4.8 minutes. A certain make of tumble dryer has a drying-time with mean 61.1 minutes and standard deviation 6.3 minutes. Both times are normally distributed and are independent of each other. Find the probability that a randomly chosen wash-time differs by more than 3 minutes from a randomly chosen drying-time.
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Hardest question Challenging +1.2 »
6 The weight, \(X\) grams, of a dressed pheasant may be modelled by a normal random variable with a mean of 1000 and a standard deviation of 120 . Pairs of dressed pheasants are selected for packing into boxes. The total weight of a pair, \(Y = X _ { 1 } + X _ { 2 }\) grams, may be modelled by a normal distribution with a mean of 2000 and a standard deviation of 140 .
    1. Show that \(\operatorname { Cov } \left( X _ { 1 } , X _ { 2 } \right) = - 4600\).
    2. Given that \(X _ { 1 } - X _ { 2 }\) may be assumed to be normally distributed, determine the probability that the difference between the weights of a selected pair of dressed pheasants exceeds 250 grams.
  2. The weight of a box is independent of the total weight of a pair of dressed pheasants, and is normally distributed with a mean of 500 grams and a standard deviation of 40 grams. Determine the probability that a box containing a pair of dressed pheasants weighs less than 2750 grams.
    \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-16_2486_1714_221_153}
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Confidence interval for single proportion

A question is this type if and only if it asks to calculate a confidence interval for a population proportion p from sample data.

8 Moderate -0.1
2.4% of questions
5.05d Confidence intervals: using normal distribution
Single normal population sample mean

Questions asking about the sample mean from a single normally distributed population, using the result that the sample mean of n observations from N(μ, σ²) has distribution N(μ, σ²/n).

8 Moderate -0.3
2.4% of questions
5.05a Sample mean distribution: central limit theorem
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1 A random variable \(X\) has the distribution \(\mathrm { N } ( 410,400 )\).
Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405 .
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Sum or total of normal variables

Questions asking about the sum or total of independent normal random variables (not the mean), using the result that the sum of independent normals is normal with summed means and variances.

8 Standard +0.0
2.4% of questions
5.04b Linear combinations: of normal distributions
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1 The lengths of logs are normally distributed with mean 3.5 m and standard deviation 0.12 m . Describe fully the distribution of the total length of 8 randomly chosen logs.
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Linear transformation to achieve target parameters

Questions that require finding constants a, b, c such that Y = aX + b (or similar) achieves specified mean and variance, working backwards from desired parameters.

7 Standard +0.3
2.1% of questions
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1 Exam marks, \(X\), have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula \(Y = a X + b\) so that the scaled marks, \(Y\), have mean 55 and standard deviation 6.96. Find the values of \(a\) and \(b\).
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Known variance confidence interval

Questions where the population variance or standard deviation is given/known in advance, so the normal distribution (z-values) is used directly for the confidence interval.

7 Moderate -0.2
2.1% of questions
5.05d Confidence intervals: using normal distribution
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1. The continuous random variable \(X\) has the distribution \(\mathrm { N } ( \mu , 30 )\). The mean of a random sample of 8 observations of \(X\) is 53.1. Determine a \(95 \%\) confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures.
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Sample size determination

A question is this type if and only if it asks to find the minimum or required sample size n to achieve a specified probability or confidence interval width.

6 Standard +0.5
1.8% of questions
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2 The lengths of sewing needles in travel sewing kits are distributed normally with mean \(\mu \mathrm { mm }\) and standard deviation 1.5 mm . A random sample of \(n\) needles is taken. Find the smallest value of \(n\) such that the width of a \(95 \%\) confidence interval for the population mean is at most 1 mm .
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Fixed container with random contents

Questions where a container has a fixed (non-random) weight and contains random items, requiring addition of a constant to a sum of normal random variables.

6 Standard +0.5
1.8% of questions
5.04b Linear combinations: of normal distributions
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4 A factory supplies boxes of children's bricks. Each box contains 10 randomly chosen large bricks and 20 randomly chosen small bricks. The masses, in grams, of large and small bricks have the distributions \(\mathrm { N } ( 60,1.2 )\) and \(\mathrm { N } ( 30,0.7 )\) respectively. The mass of an empty box is 8 g . Find the probability that the total weight of a box and its contents is less than 1200 g .
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All components random including container

Questions where both the container weight and contents weights are random variables following normal distributions, requiring linear combination of all random variables.

6 Standard +0.4
1.8% of questions
5.04b Linear combinations: of normal distributions
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1 A laminate consists of 4 layers of material \(C\) and 3 layers of material \(D\). The thickness of a layer of material \(C\) has a normal distribution with mean 1 mm and standard deviation 0.1 mm , and the thickness of a layer of material \(D\) has a normal distribution with mean 8 mm and standard deviation 0.2 mm . The layers are independent of one another.
  1. Find the mean and variance of the total thickness of the laminate.
  2. What total thickness is exceeded by \(1 \%\) of the laminates?
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Distribution of linear combination

Questions asking for the distribution of aX + bY or similar linear combinations of independent normal variables, using the result that aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²).

6 Standard +0.3
1.8% of questions
5.04b Linear combinations: of normal distributions
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2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\). Distribution of \(A\) : \(\_\_\_\_\) Distribution of \(B\) : \(\_\_\_\_\)
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Estimated variance confidence interval

Questions where the variance must be estimated from the sample data using s² or unbiased estimates, typically requiring t-distribution (though some may use normal approximation for large samples).

5 Standard +0.4
1.5% of questions
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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).
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Comparing two journey times

Questions asking for probabilities comparing two different journey times or routes (difference of normal variables), such as outbound versus return or one person versus another.

5 Standard +0.4
1.5% of questions
5.04b Linear combinations: of normal distributions
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6 The times, in minutes, taken to complete the two parts of a task are normally distributed with means 4.5 and 2.3 respectively and standard deviations 1.1 and 0.7 respectively.
  1. Find the probability that the total time taken for the task is less than 8.5 minutes.
  2. Find the probability that the time taken for the first part of the task is more than twice the time taken for the second part.
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Non-normal population sample mean (CLT)

Questions asking about the sample mean from a single non-normal population (binomial or other), requiring the Central Limit Theorem to approximate the sample mean distribution as normal.

4 Standard +0.5
1.2% of questions
5.05a Sample mean distribution: central limit theorem
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1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.
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Distribution of sample mean

Questions asking for the distribution of X̄ (sample mean) from a normal population, using the result that X̄ ~ N(μ, σ²/n).

4 Moderate -0.5
1.2% of questions
5.05a Sample mean distribution: central limit theorem
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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 \(\mathrm { P } ( \bar { N } > 354 )\).
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Total journey time probabilities

Questions asking for probabilities about the total time across multiple independent journey stages (sum of normal variables).

4 Standard +0.1
1.2% of questions
5.04b Linear combinations: of normal distributions
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5 The times, in months, taken by a builder to build two types of house, \(P\) and \(Q\), are represented by the independent variables \(T _ { 1 } \sim \mathrm {~N} \left( 2.2,0.4 ^ { 2 } \right)\) and \(T _ { 2 } \sim \mathrm {~N} \left( 2.8,0.5 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total time taken to build one house of each type is less than 6 months.
  2. Find the probability that the time taken to build a type \(Q\) house is more than 1.2 times the time taken to build a type \(P\) house.
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Single variable sum probability

Questions asking for the probability of a sum of independent observations of a single normal variable (e.g. X₁ + X₂ + ... + Xₙ where all Xᵢ have the same distribution).

3 Moderate -0.1
0.9% of questions
5.04b Linear combinations: of normal distributions
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1 The blood-test procedure at a clinic is that a person arrives, takes a numbered ticket and waits for that number to be called. The waiting times between the numbers called have independent normal distributions with mean 3.5 minutes and standard deviation 0.9 minutes. My ticket is number 39 and as I take my ticket number 1 is being called, so that I have to wait for 38 numbers to be called. Find the probability that I will have to wait between 120 minutes and 140 minutes.
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Two-sample z-test (known variances)

Questions requiring a hypothesis test comparing two population means where population standard deviations are explicitly given as known values, using a two-sample z-test with the normal distribution.

3 Standard +0.4
0.9% of questions
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2 Two brands of car battery, 'Invincible' and 'Excelsior', have lifetimes which are normally distributed. Invincible batteries have a mean lifetime of 5 years with standard deviation 0.7 years. Excelsior batteries have a mean lifetime of 4.5 years with standard deviation 0.5 years. Random samples of 20 Invincible batteries and 25 Excelsior batteries are selected and the sample mean lifetimes are \(\bar { X } _ { I }\) years and \(\bar { X } _ { E }\) years respectively.
  1. State the distributions of \(\bar { X } _ { I }\) and \(\bar { X } _ { E }\).
  2. Calculate \(\mathrm { P } \left( \bar { X } _ { I } - \bar { X } _ { E } \geqslant 1 \right)\).
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Confidence interval for difference of proportions

A question is this type if and only if it asks to construct a confidence interval for p₁ - p₂ from two independent samples of proportions.

2 Standard +0.3
0.6% of questions
5.05d Confidence intervals: using normal distribution
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5
An examination of 160 e-mails received by Gopal showed that 72 had attachments. An examination of 250 e-mails received by Haley showed that 102 had attachments.
Stating two necessary assumptions about the selection of e-mails, construct an approximate \(99 \%\) confidence interval for the difference between the proportion of e-mails received by Gopal that have attachments and the proportion of e-mails received by Haley that have attachments.
(8 marks)
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CI with known population variance

Questions that explicitly state the population standard deviation(s) are known (not estimated from sample), requiring use of the normal distribution rather than t-distribution for the confidence interval.

2 Challenging +1.0
0.6% of questions
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7 The random variable \(X\) has distribution \(\mathrm { N } ( \mu , 1 )\). A random sample of 4 observations of \(X\) is taken. The sample mean is denoted by \(\bar { X }\).
  1. Find the value of the constant \(a\) for which ( \(\bar { X } - a , \bar { X } + a\) ) is a \(98 \%\) confidence interval for \(\mu\). The independent random variable \(Y\) has distribution \(\mathrm { N } ( \mu , 9 )\). A random sample of 16 observations of \(Y\) is taken. The sample mean is denoted by \(\bar { Y }\).
  2. Write down the distribution of \(\bar { X } - \bar { Y }\).
  3. A \(90 \%\) confidence interval for \(\mu\) based on \(\bar { Y }\) is given by ( \(\bar { Y } - 1.234 , \bar { Y } + 1.234\) ). Find the probability that this interval does not overlap with the interval in part (i).
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Distribution of scaled variable

Questions asking for the distribution of aX (a single normal variable multiplied by a constant), using the result that aX ~ N(aμ, a²σ²).

2 Moderate -0.2
0.6% of questions
5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions
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4 The volume, in millilitres, of a small cup of coffee has the distribution \(\mathrm { N } ( 103.4,10.2 )\). The volume of a large cup of coffee is 1.5 times the volume of a small cup of coffee.
  1. Find the mean and standard deviation of the volume of a large cup of coffee.
  2. Find the probability that the total volume of a randomly chosen small cup of coffee and a randomly chosen large cup of coffee is greater than 250 ml .
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Single period profit/loss calculation

Questions involving profit, loss, or cost for a single time period (one day, one month, one year) where the financial quantity is a linear combination of random variables.

2 Standard +0.3
0.6% of questions
5.04b Linear combinations: of normal distributions
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6 A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)\) and \(\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)\) respectively. Dried yeast costs \(\\) 13.50\( per kilogram and flour costs \)\\( 0.90\) per kilogram. For making one batch of bread the total of all other costs is \(\\) 55\(. The factory sells each batch of bread for \)\\( 200\). Find the probability that the profit made on one randomly chosen batch of bread is greater than \(\\) 40$. [7]
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Percentage/proportion exceeding threshold

A question is this type if and only if it asks to estimate what percentage or proportion of a population exceeds or falls below a specified value using normal distribution.

1 Standard +0.3
0.3% of questions
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  1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml
Given that 15\% of bottles contain less than 24.63 ml
  1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
  2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\) The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml
  3. Test Hannah's belief at the \(5 \%\) level of significance. You should state your hypotheses clearly.
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Multiple period profit/loss calculation

Questions involving profit, loss, or cost aggregated over multiple independent time periods (multiple days, weeks, months) requiring use of properties of sums of independent normal variables.

1 Standard +0.3
0.3% of questions
5.04a Linear combinations: E(aX+bY), Var(aX+bY)
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4 The mass, in tonnes, of steel produced per day at a factory is normally distributed with mean 65.2 and standard deviation 3.6. It can be assumed that the mass of steel produced each day is independent of other days. The factory makes \(\\) 50$ profit on each tonne of steel produced. Find the probability that the total profit made in a randomly chosen 7-day week is less than \(\\) 22000$.
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Paired t-test

Questions requiring a hypothesis test on paired or matched data where the same subjects are measured twice or subjects are naturally paired, testing the mean of differences using a paired t-test.

0
0.0% of questions
Direct comparison with k=1 (same variable)

Questions where two independent observations from the same distribution are compared (e.g., P(X1 > X2)) or where k=1 and both variables have the same distribution, requiring distribution of X1 - X2.

0
0.0% of questions
Unclassified

Questions not yet assigned to a type.

16
4.8% of questions
Show 16 unclassified »
5 The masses, in grams, of large and small packets of Maxwheat cereal have the independent distributions \(\mathrm { N } \left( 410.0,3.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 206.0,3.7 ^ { 2 } \right)\) respectively.
  1. Find the probability that a randomly chosen large packet has a mass that is more than double the mass of a randomly chosen small packet.
    The packets are placed in boxes. The boxes are identical in appearance. \(60 \%\) of the boxes contain exactly 10 randomly chosen large packets. 40\% of the boxes contain exactly 20 randomly chosen small packets.
  2. Find the probability that a randomly chosen box contains packets with a total mass of more than 4080 grams.
3 The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions \(\mathrm { N } \left( 40,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 12,0.7 ^ { 2 } \right)\) respectively.
  1. Find the probability that the total mass of 6 randomly chosen large sacks of flour is more than 245 kg .
  2. Find the probability that the mass of a randomly chosen large sack of flour is less than 4 times the mass of a randomly chosen small sack of flour.
5 The volumes, in litres, of juice in large and small bottles have the distributions \(\mathrm { N } ( 5.10,0.0102 )\) and \(\mathrm { N } ( 2.51,0.0036 )\) respectively.
  1. Find the probability that the total volume of juice in 3 randomly chosen large bottles and 4 randomly chosen small bottles is less than 25.5 litres.
  2. Find the probability that the volume of juice in a randomly chosen large bottle is at least twice the volume of juice in a randomly chosen small bottle.
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.
4 Eezimix flour is sold in small bags of weight \(S\) grams, where \(S \sim \mathrm {~N} \left( 502.1,0.31 ^ { 2 } \right)\). It is also sold in large bags of weight \(L\) grams, where \(L \sim \mathrm {~N} \left( 1004.9,0.58 ^ { 2 } \right)\).
  1. Find the probability that a randomly chosen large bag weighs at least 1 gram more than two randomly chosen small bags.
  2. Find the probability that a randomly chosen large bag weighs less than twice the weight of a randomly chosen small bag.
5. The weights, in kg , of cars may be assumed to follow the normal distribution \(\mathrm { N } \left( 1000,250 ^ { 2 } \right)\). The weights, in kg , of lorries may be assumed to follow the normal distribution \(\mathrm { N } \left( 2800,650 ^ { 2 } \right)\). A lorry and a car are chosen at random.
  1. Find the probability that the lorry weighs more than 3 times the weight of the car. A ferry carries vehicles across a river. The ferry is designed to carry a maximum weight of 20000 kg .
  2. One morning, 8 cars and 3 lorries drive on to the ferry. Find the probability that their total weight will exceed the recommended maximum weight of 20000 kg .
  3. State a necessary assumption needed for the calculation in part (b).
7 A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic. The weights, \(W \mathrm {~kg}\), of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg . The weights, \(X \mathrm {~kg}\), of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg . The random variables \(W\) and \(X\) are independent.
  1. Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy. The manufacturer packs \(n\) of these wooden toys and \(2 n\) of these plastic toys into the same container. The maximum weight the container can hold is 252 kg . The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.
  2. Calculate the value of \(n\).
  1. Scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.6,0.6 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution N(4.8, 0.32). The random variables \(L\) and \(S\) are independent.
A long pole and a short pole are selected at random.
  1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Show your working clearly. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
  2. Find the distribution of \(T\).
  3. Find \(\mathrm { P } ( | L - T | < 0.2 )\)
11 The thickness of a randomly chosen paperback book is \(P \mathrm {~cm}\) and the thickness of a randomly chosen hardback is \(H \mathrm {~cm}\), where \(P\) and \(H\) have distributions \(\mathrm { N } ( 2.0,0.75 )\) and \(\mathrm { N } ( 5.0,2.25 )\) respectively. When more than one book is selected, any book is selected independently of all other books.
  1. Calculate the probability that a randomly chosen hardback is more than 1 cm thicker than a randomly chosen paperback.
  2. Calculate the probability that 2 paperbacks and 4 hardbacks, randomly chosen, have a combined thickness of less than 20 cm .
  3. Find the probability that a randomly chosen hardback is more than twice the thickness of a randomly chosen paperback.
Answer only one of the following two alternatives. EITHER \includegraphics{figure_10a} An object is formed by attaching a thin uniform rod \(PQ\) to a uniform rectangular lamina \(ABCD\). The lamina has mass \(m\), and \(AB = DC = 6a\), \(BC = AD = 3a\). The rod has mass \(M\) and length \(3a\). The end \(P\) of the rod is attached to the mid-point of \(AB\). The rod is perpendicular to \(AB\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l_1\), through \(Q\) and perpendicular to the plane of the lamina, is \(3(8m + M)a^2\). [4] Show that the moment of inertia of the object about a smooth horizontal axis \(l_2\), through the mid-point of \(PQ\) and perpendicular to the plane of the lamina, is \(\frac{3}{4}(17m + M)a^2\). [2] Find expressions for the periods of small oscillations of the object about the axes \(l_1\) and \(l_2\), and verify that these periods are equal when \(m = M\). [8] OR A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x\) kg. The data are summarised as follows. $$\Sigma x = 42.0 \qquad \Sigma x^2 = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y\) kg. The data are summarised as follows. $$\Sigma y = 57.6 \qquad \Sigma y^2 = 281.5$$ Test, at the 5% significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. [9] A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\)'s Crown plants. Test, at the 5% significance level, whether Farmer \(A\)'s claim is justified. [5]
The masses, in kilograms, of large and small sacks of flour have the distributions \(\text{N}(55, 3^2)\) and \(\text{N}(27, 2.5^2)\) respectively.
  1. Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340 kg. Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour. [5]
  2. Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour. [5]
Bottles of wine are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.3 kg and standard deviation 0.06 kg. The weights of the empty racks are normally distributed with mean 2 kg and standard deviation 0.3 kg.
  1. Find the probability that the total weight of a full rack of 12 bottles of wine is between 17 kg and 18 kg. [5]
  2. Two bottles of wine are chosen at random. Find the probability that they differ in weight by more than 0.05 kg. [5]
A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg
  1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg [6]
Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
  1. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg [5]
An examiner believes that once she has marked the first 20 papers the time it takes her to mark one paper for a particular exam follows a Normal distribution. Having already marked more than 20 papers for each of the \(P1\), \(M1\) and \(S1\) modules set one summer, the mean and standard deviation, in seconds, of the time it takes her to mark a paper for each module are as shown in the table below.
MeanStandard Deviation
\(P1\)25217
\(M1\)31442
\(S1\)28429
  1. Find the probability that the difference in the time it takes her to mark two randomly chosen \(P1\) papers is less than 5 seconds. [6]
  2. Find the probability that it takes her less than 10 hours to mark 45 \(M1\) and 80 \(S1\) papers. [7]
An organic farm produces eggs which it sells through a local shop. The weight of the eggs produced on the farm are normally distributed with a mean of 55 grams and a standard deviation of 3.9 grams.
  1. Find the probability that two of the farm's eggs chosen at random differ in weight by more than 4 grams. [5]
The farm sells boxes of six eggs selected at random. The weight of the boxes used are normally distributed with a mean of 28 grams and a standard deviation of 1.2 grams.
  1. Find the probability that a randomly chosen box with six eggs in weighs less than 350 grams. [6]
The mass \(J\) kg of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass \(K\) kg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04.
  1. Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg. [3]
  2. Find the probability that the mass of one bag of King Edward potatoes is more than 75\% of the mass of one bag of Jersey potatoes. [3]