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

2 The weight, \(X\) grams, of a particular variety of orange is normally distributed with mean 205 and standard deviation 25.

1. Determine the probability that the weight of an orange is:
   1. less than 250 grams;
   2. between 200 grams and 250 grams.
2. A wholesaler decides to grade such oranges by weight. He decides that the smallest 30 per cent should be graded as small, the largest 20 per cent graded as large, and the remainder graded as medium. Determine, to one decimal place, the maximum weight of an orange graded as:
   1. small;
   2. medium.
3. The weight, \(Y\) grams, of a second variety of orange is normally distributed with mean 175. Given that 90 per cent of these oranges weigh less than 200 grams, calculate the standard deviation of their weights.
   (4 marks)

2 The heights of sunflowers may be assumed to be normally distributed with a mean of 185 cm and a standard deviation of 10 cm .

1. Determine the probability that the height of a randomly selected sunflower:
   1. is less than 200 cm ;
   2. is more than 175 cm ;
   3. is between 175 cm and 200 cm .
2. Determine the probability that the mean height of a random sample of 4 sunflowers is more than 190 cm .

4 The weights of packets of sultanas may be assumed to be normally distributed with a standard deviation of 6 grams. The weights of a random sample of 10 packets were as follows: \(\begin{array} { l l l l l l l l l l } 498 & 496 & 499 & 511 & 503 & 505 & 510 & 509 & 513 & 508 \end{array}\)

1. 1. Construct a \(99 \%\) confidence interval for the mean weight of packets of sultanas, giving the limits to one decimal place.
   2. State why, in calculating your confidence interval, use of the Central Limit Theorem was not necessary.
   3. On each packet it states 'Contents 500 grams'. Comment on this statement using both the given sample and your confidence interval.
2. Given that the mean weight of all packets of sultanas is 500 grams, state the probability that a 99\% confidence interval for the mean, calculated from a random sample of packets, will not contain 500 grams.

2 The length of aluminium baking foil on a roll may be modelled by a normal distribution with mean 91 metres and standard deviation 0.8 metres.

1. Determine the probability that the length of foil on a particular roll is:
   1. less than 90 metres;
   2. not exactly 90 metres;
   3. between 91 metres and 92.5 metres.
2. The length of cling film on a roll may also be modelled by a normal distribution but with mean 153 metres and standard deviation \(\sigma\) metres. It is required that \(1 \%\) of rolls of cling film should have a length less than 150 metres.
   Find the value of \(\sigma\) that is needed to satisfy this requirement.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-04_1526_1714_1181_153}

7

1. A greengrocer displays apples in trays. Each customer selects the apples he or she wishes to buy and puts them into a bag. Records show that the weight of such bags of apples may be modelled by a normal distribution with mean 1.16 kg and standard deviation 0.43 kg . Determine the probability that the mean weight of a random sample of 10 such bags of apples exceeds 1.25 kg .
2. The greengrocer also displays pears in trays. Each customer selects the pears he or she wishes to buy and puts them into a bag. A random sample of 40 such bags of pears had a mean weight of 0.86 kg and a standard deviation of 0.65 kg .
   1. Construct a \(\mathbf { 9 6 \% }\) confidence interval for the mean weight of a bag of pears.
   2. Hence comment on a claim that customers wish to buy, on average, a greater weight of apples than of pears.
      [0pt] [2 marks]

5

1. Wooden lawn edging is supplied in 1.8 m length rolls. The actual length, \(X\) metres, of a roll may be modelled by a normal distribution with mean 1.81 and standard deviation 0.08 . Determine the probability that a randomly selected roll has length:
   1. less than 1.90 m ;
   2. greater than 1.85 m ;
   3. between 1.81 m and 1.85 m .
2. Plastic lawn edging is supplied in 9 m length rolls. The actual length, \(Y\) metres, of a roll may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). An analysis of a batch of rolls, selected at random, showed that $$\mathrm { P } ( Y < 9.25 ) = 0.88$$
   1. Use this probability to find the value of \(z\) such that $$9.25 - \mu = z \times \sigma$$ where \(z\) is a value of \(Z \sim \mathrm {~N} ( 0,1 )\).
   2. Given also that $$\mathrm { P } ( Y > 8.75 ) = 0.975$$ find values for \(\mu\) and \(\sigma\).

5 In an investment model the increase, \(Y \%\), in the value of an investment in one year is modelled as a continuous random variable with the distribution \(\mathrm { N } \left( \mu , \frac { 1 } { 4 } \mu ^ { 2 } \right)\). The value of \(\mu\) depends on the type of investment chosen.

1. Find \(\mathrm { P } ( Y < 0 )\), showing that it is independent of the value of \(\mu\).
2. Given that \(\mu = 6\), find the probability that \(Y < 9\) in each of three randomly chosen years.
3. Explain why the calculation in part (ii) might not be valid if applied to three consecutive years.

9 The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-08_842_1651_495_207} One of the 150 plants is chosen at random, and its height, \(X \mathrm {~cm}\), is noted.

1. Show that \(\mathrm { P } ( 20 < X < 30 ) = 0.147\), correct to 3 significant figures. Sam suggests that the distribution of \(X\) can be well modelled by the distribution \(\mathrm { N } ( 40,100 )\).
2. 1. Give a brief justification for the use of the normal distribution in this context.
   2. Give a brief justification for the choice of the parameter values 40 and 100 .
3. Use Sam's model to find \(\mathrm { P } ( 20 < X < 30 )\). Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution \(\mathrm { N } \left( m , s ^ { 2 } \right)\) as her model.
4. Use Nina's model to find \(\mathrm { P } ( 20 < X < 30 )\).
5. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution.

1. The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.

Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.

1. Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
2. Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
3. Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.

1. A machine cuts strips of metal to length \(L \mathrm {~cm}\), where \(L\) is normally distributed with standard deviation 0.5 cm .

Strips with length either less than 49 cm or greater than 50.75 cm cannot be used.
Given that 2.5\% of the cut lengths exceed 50.98 cm ,

1. find the probability that a randomly chosen strip of metal can be used. Ten strips of metal are selected at random.
2. Find the probability fewer than 4 of these strips cannot be used. A second machine cuts strips of metal of length \(X \mathrm {~cm}\), where \(X\) is normally distributed with standard deviation 0.6 cm A random sample of 15 strips cut by this second machine was found to have a mean length of 50.4 cm
3. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test whether or not the mean length of all the strips, cut by the second machine, is greater than 50.1 cm

5. The lifetimes of batteries sold by company \(X\) are normally distributed, with mean 150 hours and standard deviation 25 hours. A box contains 12 batteries from company \(X\).

1. Find the expected number of these batteries that have a lifetime of more than 160 hours. The lifetimes of batteries sold by company \(Y\) are normally distributed, with mean 160 hours and \(80 \%\) of these batteries have a lifetime of less than 180 hours.
2. Find the standard deviation of the lifetimes of batteries from company \(Y\). Both companies sell their batteries for the same price.
3. State which company you would recommend. Give reasons for your answer.

9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12 9

1. The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to \(p\) Lianne will only buy the battery if \(p\) is more than 0.4
   Determine whether Lianne will buy the battery.
   [0pt] [2 marks]
   9
2. A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427} Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
   \includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}

5 A particular brand of pasta is sold in bags of two different sizes. The mass of pasta in the large bags is advertised as being 1500 g ; in fact it is Normally distributed with mean 1515 g and standard deviation 4.7 g . The mass of pasta in the small bags is advertised as being 500 g ; in fact it is Normally distributed with mean 508 g and standard deviation 3.3 g .

1. Find the probability that the total mass of pasta in 5 randomly selected small bags is less than 2550 g .
2. Find the probability that the mass of pasta in a randomly selected large bag is greater than three times the mass of pasta in a randomly selected small bag.

3. A string of length 60 cm is cut a random point.

1. Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
2. The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than \(100 \mathrm {~cm} ^ { 2 }\).

4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects \(2 p\) coins that are less than 6.12 grams or greater than 8.12 grams.

1. The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Weight of UK two pence coins} \includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
   \end{figure} Explain why the weight of 2 p coins can be modelled using a normal distribution.
2. Assume the distribution of the weight of \(2 p\) coins is normally distributed. Calculate the proportion of \(2 p\) coins that are rejected by this brand of coin counting machine.
3. A manager suspects that a large batch of \(2 p\) coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.

   Summary statistics
   Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
   6.89	0.296	6.45	6.63	6.88	7.08	7.48

   i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.
   ii) Assuming the population standard deviation is 0.357 grams, test at the \(1 \%\) significance level whether the mean weight of the \(2 p\) coins in this batch is less than 7.12 grams.

1. Rachel records the times taken, in minutes, to cycle into town from her house on a random sample of 10 days. Her results are shown below.

$$\begin{array} { l l l l l l l l l l } 15 \cdot 5 & 14 \cdot 9 & 16 \cdot 2 & 17 \cdot 3 & 14 \cdot 8 & 14 \cdot 2 & 16 \cdot 0 & 14 \cdot 2 & 15 \cdot 5 & 15 \cdot 1 \end{array}$$ Assuming that these data come from a normal distribution with mean \(\mu\) and variance \(0 \cdot 9\), calculate a \(90 \%\) confidence interval for \(\mu\).

2. Geraint is a beekeeper. The amounts of honey, \(X \mathrm {~kg}\), that he collects annually, from each hive are modelled by the normal distribution \(\mathrm { N } \left( 15,5 ^ { 2 } \right)\). At location \(A\), Geraint has three hives and at location \(B\) he has five hives. You may assume that the amounts of honey collected from the eight hives are independent of each other.

1. 1. Find the probability that Geraint collects more than 14 kg of honey from the first hive at location \(A\).
   2. Find the probability that he collects more than 14 kg of honey from exactly two out of the three hives at location \(A\).
2. Find the probability that the total amount of honey that Geraint collects from all eight hives is more than 160 kg .
3. Find the probability that Geraint collects at least twice as much honey from location B as from location A.

4. The Department of Health recommends that adults aged 18 to 65 should take part in at least 150 minutes of aerobic exercise per week. The results of a survey show that 940 out of 2000 randomly selected adults aged 18 to 65 in Wales take part in at least 150 minutes of aerobic exercise per week.

1. Calculate an approximate \(95 \%\) confidence interval for the proportion of adults aged 18 to 65 in Wales who take part in at least 150 minutes of aerobic exercise per week.
2. Give two reasons why the interval is approximate.
3. Suppose that a \(99 \%\) confidence interval is required, and that the width of the interval is to be no greater than \(0 \cdot 04\). Estimate the minimum additional number of adults to be surveyed to satisfy this requirement.

7. \includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral \(A B C D\), where \(\widehat { B A D } = \alpha , \widehat { B C D } = \beta\) and \(\alpha + \beta = 180 ^ { \circ }\). These angles are measured.
The random variables \(X\) and \(Y\) denote the measured values, in degrees, of \(\widehat { B A D }\) and \(\widehat { B C D }\) respectively. You are given that \(X\) and \(Y\) are independently normally distributed with standard deviation \(\sigma\) and means \(\alpha\) and \(\beta\) respectively.

1. Calculate, correct to two decimal places, the probability that \(X + Y\) will differ from \(180 ^ { \circ }\) by less than \(\sigma\).
2. Show that \(T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )\) is an unbiased estimator for \(\alpha\) and verify that it is a better estimator than \(X\) for \(\alpha\).
3. Now consider \(T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)\).
   1. Show that \(T _ { 2 }\) is an unbiased estimator for \(\alpha\) for all values of \(\lambda\).
   2. Find \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(\sigma\).
   3. Hence determine the value of \(\lambda\) which gives the best unbiased estimator for \(\alpha\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The distances, \(m\) miles, a motorbike travels on a full tank of petrol can be modelled by a normal distribution with mean 170 miles and standard deviation 16 miles.

1. Find the probability that, on a randomly selected journey, the motorbike could travel at least 190 miles on a full tank of petrol. The probability that, on a randomly selected journey, the motorbike could travel at least \(d\) miles on a full tank of petrol is 0.9
2. Find the value of \(d\)

1. An orchard produces apples.

The weights, \(A\) grams, of its apples are normally distributed with mean \(\mu\) grams and standard deviation \(\sigma\) grams. It is known that $$\mathrm { P } ( A < 162 ) = 0.1 \text { and } \mathrm { P } ( 162 < A < 175 ) = 0.7508$$

1. Calculate the value of \(\mu\) and the value of \(\sigma\) A second orchard also produces apples.
   The weights, \(B\) grams, of its apples have distribution \(B \sim N \left( 215,10 ^ { 2 } \right)\) An outlier is a value that is
   greater than \(\mathrm { Q } _ { 3 } + 1.5 \times \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) or smaller than \(\mathrm { Q } _ { 1 } - 1.5 \times \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) An apple is selected at random from this second orchard.
   Using \(\mathrm { Q } _ { 3 } = 221.74\) grams,
2. find the probability that this apple is an outlier.

3 The lengths of snakes on a tropical island were measured and found to be normally distributed with a mean of 160 cm and a standard deviation of 6 cm . Find the probability that a randomly selected snake has a length of less than 170 cm .

4 A tomato grower grows just one variety of tomatoes. The weights of these tomatoes are found to be normally distributed with a mean of 85.1 grams and a standard deviation of 3.4 grams.

1. Find the probability that a randomly chosen tomato of this variety weighs less than 80 grams.
2. The grower puts the tomatoes in packs of 6 . Find the probability that, in a randomly chosen pack of 6 , at most one tomato weighs less than 80 grams.
3. The grower supplies consignments of 250 packs of these tomatoes to a retailer. For a randomly chosen consignment, find the expected number of packs having more than one tomato weighing less than 80 grams.

4 It is given that \(X\) and \(Y\) are independent random variables with distributions \(\mathrm { N } \left( \mu _ { x } , \sigma _ { x } ^ { 2 } \right)\) and \(\mathrm { N } \left( \mu _ { y } , \sigma _ { y } ^ { 2 } \right)\) respectively, and that \(W\) is a random variable such that \(W = X + Y\).

1. Use moment generating functions to show that the distribution of \(W\) is \(\mathrm { N } \left( \mu _ { x } + \mu _ { y } , \sigma _ { x } ^ { 2 } + \sigma _ { y } ^ { 2 } \right)\).
2. State the distribution of \(X - Y\). The diameters of the central poles of one brand of rotary clothes lines are normally distributed with mean 3.75 cm and variance \(0.000125 \mathrm {~cm} ^ { 2 }\). The diameters of the cylindrical tubes, into which the central poles fit, are normally distributed with mean 3.85 cm and variance \(0.0001 \mathrm {~cm} ^ { 2 }\). Poles and tubes are chosen at random. The 'clearance' between a tube and a pole is the diameter of the tube minus the diameter of the pole.
3. Find the probability that a pole and tube have a clearance between 0.08 cm and 0.13 cm .
4. Given that a pole and tube have a clearance between 0.08 cm and 0.13 cm , find the probability that the clearance is between 0.11 cm and 0.125 cm .

3 Jack's journey time, in minutes, to work each morning is modelled by the normal distribution \(\mathrm { N } \left( 43.2,6.3 ^ { 2 } \right)\).

1. If Jack leaves home at 0810 , find the probability that he arrives at work by 0900 .
2. Find the time by which Jack should leave home in order to be at least \(95 \%\) certain that he arrives at work by 0900 .