2.04e Normal distribution: as model N(mu, sigma^2)

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Pre-U Pre-U 9795/2 2016 June Q2
4 marks Standard +0.8
2 The mass in grams of a pre-cut piece of Brie cheese is a random variable with the distribution \(\mathrm { N } ( 150,1200 )\). Brie costs 80 p per 100 g .
  1. Find the probability that a randomly chosen piece of Brie costs more than \(\pounds 1.40\). The mass in grams of a pre-cut piece of Stilton cheese is an independent random variable with the distribution \(\mathrm { N } ( 180,1500 )\).
  2. Find the probability that the total mass of four randomly chosen pieces of Brie is less than the total mass of three randomly chosen pieces of Stilton.
Pre-U Pre-U 9794/3 2016 Specimen Q1
5 marks Moderate -0.8
1 The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes
  1. more than 27 minutes,
  2. between 20 and 25 minutes.
Pre-U Pre-U 9794/3 2016 Specimen Q5
11 marks Moderate -0.8
5 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.
  1. Construct a probability distribution table for \(X\).
  2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
  3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
  4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.
Pre-U Pre-U 9794/3 2018 June Q5
9 marks Standard +0.3
5 A soft drinks company has an automated bottling machine that fills 500 ml bottles with soft drink. The contents of the bottles are measured during a check on the machine. In the check, \(5 \%\) of the bottles contain more than 500 ml and \(2.5 \%\) contain less than 495 ml . It is given that the amount of drink dispensed per bottle is normally distributed.
  1. Find the mean and standard deviation of the amount of drink dispensed per bottle, giving your answers to 4 significant figures.
  2. It is subsequently found that the measurements of volume made in the checking process are all 3 ml below their true value. Using a corrected distribution, find the probability that a bottle chosen at random contains more than 500 ml of the drink.
Pre-U Pre-U 9794/1 Specimen Q14
14 marks Moderate -0.3
14 The maximum pressure exerted by the blood on the arteries in a population of elderly male patients may be modelled by a random variable having a normal distribution with a mean of 150 and standard deviation 15, measured in suitable units.
  1. Find the probability that the maximum pressure for a randomly chosen patient is more than 160.
  2. If the maximum pressure is found to be \(t\) or more, the patient must be referred to a consultant. If \(5 \%\) of the patients are referred to a consultant, find the value of \(t\).
  3. Find the percentage of patients whose maximum pressure is between 130 and 160 . The probability that a randomly chosen patient attending a doctor's surgery has their blood pressure measured is 0.4 .
  4. Find the probability that of 18 people attending a doctor's surgery more than 8 have their blood pressure measured, assuming that each measurement is random and independent of any other.
  5. If 450 patients visited the surgery in a week, find the expected number of patients whose blood pressure would be measured.
Pre-U Pre-U 9794/1 Specimen Q10
9 marks Standard +0.3
10 Cheeky Cola is sold in bottles of two sizes, small and large. For each size, the content of a randomly chosen bottle is normally distributed with mean and standard deviation, in litres, as given in the table.
MeanStandard deviation
Small bottle0.50.01
Large bottle1.5\(x\)
  1. Find the probability that a randomly chosen small bottle contains more than 0.51 litres.
  2. Find \(x\) if the probability that a randomly chosen large bottle contains less than 1.45 litres is 0.1 . The manufacturer introduces a new size of bottle of Cheeky Cola, called the mega bottle. It is found that the probabilities that a randomly chosen mega bottle contains less than 2.97 litres or more than 3.05 litres are both 0.05 .
  3. Assuming that the contents of the mega bottle are normally distributed, find the mean and variance of the distribution.
Pre-U Pre-U 9795/2 Specimen Q7
1 marks Standard +0.8
7 The time taken for me to walk from my house to the bus stop has a normal distribution with mean 10 minutes and standard deviation 1.5 minutes. The arrival time of the bus is normally distributed with mean 0900 and standard deviation 1 minute. If the bus arrives early it does not wait. I leave home at 0845 . Find, correct to 3 decimal places, the probability that I catch the bus.
CAIE S1 2023 March Q6
11 marks Standard +0.3
In a cycling event the times taken to complete a course are modelled by a normal distribution with mean 62.3 minutes and standard deviation 8.4 minutes.
  1. Find the probability that a randomly chosen cyclist has a time less than 74 minutes. [2]
  2. Find the probability that 4 randomly chosen cyclists all have times between 50 and 74 minutes. [4]
In a different cycling event, the times can also be modelled by a normal distribution. 23\% of the cyclists have times less than 36 minutes and 10\% of the cyclists have times greater than 54 minutes.
  1. Find estimates for the mean and standard deviation of this distribution. [5]
CAIE S1 2002 June Q4
7 marks Moderate -0.8
  1. In a spot check of the speeds \(x \text{ km h}^{-1}\) of 30 cars on a motorway, the data were summarised by \(\Sigma(x - 110) = -47.2\) and \(\Sigma(x - 110)^2 = 5460\). Calculate the mean and standard deviation of these speeds. [4]
  2. On another day the mean speed of cars on the motorway was found to be \(107.6 \text{ km h}^{-1}\) and the standard deviation was \(13.8 \text{ km h}^{-1}\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \text{ km h}^{-1}\), find what proportion of cars exceed the speed limit. [3]
CAIE S1 2002 June Q6
8 marks Standard +0.3
  1. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), \(\text{P}(X > 3.6) = 0.5\) and \(\text{P}(X > 2.8) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\). [4]
  2. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8. [4]
CAIE S1 2010 June Q2
5 marks Moderate -0.8
The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm.
  1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm. [2]
  2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm. [3]
CAIE S1 2010 June Q4
6 marks Standard +0.3
The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
  1. Given that \(5\sigma = 3\mu\), find \(\mathrm{P}(X < 2\mu)\). [3]
  2. With a different relationship between \(\mu\) and \(\sigma\), it is given that \(\mathrm{P}(X < \frac{4\mu}{3}) = 0.8524\). Express \(\mu\) in terms of \(\sigma\). [3]
CAIE S1 2015 June Q7
12 marks Moderate -0.3
  1. Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with mean 35.2 and standard deviation 4.7.
    1. Find the expected number of days during a year (52 weeks) for which Zak takes less than 30 minutes for his run. [4]
    2. The probability that Zak's time is between 35.2 minutes and \(t\) minutes, where \(t > 35.2\), is 0.148. Find the value of \(t\). [3]
  2. The random variable \(X\) has the distribution \(\text{N}(\mu, \sigma^2)\). It is given that \(\text{P}(X < 7) = 0.2119\) and \(\text{P}(X < 10) = 0.6700\). Find the values of \(\mu\) and \(\sigma\). [5]
CAIE S1 2014 November Q5
8 marks Moderate -0.8
  1. The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
    1. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night. [2]
    2. Find the value of \(q\) such that P\((X < q) = 0.75\). [3]
  2. The random variable \(Y\) has the distribution N\((\mu, \sigma^2)\), where \(2\sigma = 3\mu\) and \(\mu \neq 0\). Find P\((Y > 4\mu)\). [3]
CAIE S2 2002 November Q7
10 marks Standard +0.3
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]
Edexcel S1 2023 June Q7
14 marks Standard +0.3
A machine squeezes apples to extract their juice. The volume of juice, \(J\) ml, extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to
    1. show that P\((J > 510) = 0.3446\) [2]
    2. calculate the value of \(d\) such that P\((J > d) = 0.9192\) [3]
Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
  1. Calculate the probability that each of the 5 bags of apples produce less than 510ml of juice. [2]
Following adjustments to the machine, the volume of juice, \(R\) ml, extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that P\((R < r) = 0.15\) and P\((R > 3r - 800) = 0.005\)
  1. find the value of \(r\) and the value of \(k\) [7]
Edexcel S1 2002 January Q5
11 marks Standard +0.3
The duration of the pregnancy of a certain breed of cow is normally distributed with mean \(\mu\) days and standard deviation \(\sigma\) days. Only 2.5\% of all pregnancies are shorter than 235 days and 15\% are longer than 286 days.
  1. Show that \(\mu - 235 = 1.96\sigma\). [2]
  2. Obtain a second equation in \(\mu\) and \(\sigma\). [3]
  3. Find the value of \(\mu\) and the value of \(\sigma\). [4]
  4. Find the values between which the middle 68.3\% of pregnancies lie. [2]
Edexcel S1 2010 January Q7
13 marks Standard +0.3
The heights of a population of women are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 30% of the women are taller than 172 cm and 5% are shorter than 154 cm.
  1. Sketch a diagram to show the distribution of heights represented by this information. [3]
  2. Show that \(\mu = 154 + 1.6449\sigma\). [3]
  3. Obtain a second equation and hence find the value of \(\mu\) and the value of \(\sigma\). [4]
A woman is chosen at random from the population.
  1. Find the probability that she is taller than 160 cm. [3]
Edexcel S1 2011 June Q2
5 marks Moderate -0.8
The random variable \(X \sim \text{N}(\mu, 5^2)\) and \(\text{P}(X < 23) = 0.9192\)
  1. Find the value of \(\mu\). [4]
  2. Write down the value of \(\text{P}(\mu < X < 23)\). [1]
Edexcel S1 2011 June Q4
7 marks Moderate -0.8
Past records show that the times, in seconds, taken to run 100 m by children at a school can be modelled by a normal distribution with a mean of 16.12 and a standard deviation of 1.60 A child from the school is selected at random.
  1. Find the probability that this child runs 100 m in less than 15 s. [3]
On sports day the school awards certificates to the fastest 30\% of the children in the 100 m race.
  1. Estimate, to 2 decimal places, the slowest time taken to run 100 m for which a child will be awarded a certificate. [4]
Edexcel S1 2011 June Q5
11 marks Moderate -0.8
A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarised in the table below.
TimeMid-point, \(x\)Frequency, \(f\)
2 - 852
9 - 127
13 - 15145
16 - 18178
19 - 2220.54
23 - 3026.54
(You may use \(\sum fx^2 = 8603.75\))
  1. Write down the mid-point for the 9 - 12 interval. [1]
  2. Use linear interpolation to estimate the median time taken by the students. [2]
  3. Estimate the mean and standard deviation of the times taken by the students. [5]
The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.
  1. Give a reason to support the use of a normal distribution in this case. [1]
On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found \(Q_1 = 8.5 \quad Q_2 = 13.0 \quad Q_3 = 21.0\)
  1. Describe, giving a reason, the skewness of the times on this occasion. [2]
Edexcel S1 2002 November Q1
4 marks Easy -1.8
  1. Explain briefly why statistical models are used when attempting to solve real-world problems. [2]
  2. Write down the name of the distribution you would recommend as a suitable model for each of the following situations.
    1. The weight of marmalade in a jar.
    2. The number on the uppermost face of a fair die after it has been rolled.
    [2]
Edexcel S1 2002 November Q4
11 marks Moderate -0.3
Strips of metal are cut to length \(L\) cm, where \(L \sim N(\mu, 0.5^2)\).
  1. Given that 2.5\% of the cut lengths exceed 50.98 cm, show that \(\mu = 50\). [5]
  2. Find \(P(49.25 < L < 50.75)\). [4]
Those strips with length either less than 49.25 cm or greater than 50.75 cm cannot be used. Two strips of metal are selected at random.
  1. Find the probability that both strips cannot be used. [2]
Edexcel S1 Specimen Q1
4 marks Easy -1.8
  1. Explain what you understand by a statistical model. [2]
  2. Write down a random variable which could be modelled by
    1. a discrete uniform distribution,
    2. a normal distribution.
    [2]
Edexcel S1 Specimen Q2
9 marks Standard +0.8
A group of students believes that the time taken to travel to college, \(T\) minutes, can be assumed to be normally distributed. Within the college 5\% of students take at least 55 minutes to travel to college and 0.1\% take less than 10 minutes. Find the mean and standard deviation of \(T\). [9]