2.04f Find normal probabilities: Z transformation

6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.

1. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
2. \(20 \%\) of employees take longer than \(t\) minutes to complete the task. Find the value of \(t\).
3. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.

Jia writes a computer program that randomly generates values from a normal distribution. He sets the mean as 40 and the standard deviation as 2.4

1. Find the probability that a particular value generated by the computer program is less than 37 Jia changes the mean to \(m\) but leaves the standard deviation as 2.4
   The computer program then randomly generates 2 independent values from this normal distribution. The probability that both of these values are greater than 32 is 0.16
2. Find the value of \(m\), giving your answer to 2 decimal places. Jia now changes the mean to 4 and the standard deviation to 8
   The computer program then randomly generates 5 independent values from this normal distribution.
3. Find the probability that at least one of these values is negative.

At a school athletics day, the distances, in metres, achieved by students in the long jump are modelled by the normal distribution with mean 3.3 m and standard deviation 0.6 m

1. Find an estimate for the proportion of students who jump less than 2.5 m The long jump competition consists of 2 jumps. All the students can take part in the first jump and the \(40 \%\) who jump the greatest distance in their first jump qualify for the second jump.
2. Find an estimate for the minimum distance achieved in the first jump in order to qualify for the second jump.
   Give your answer correct to 4 significant figures.
3. Find an estimate for the median distance achieved in the first jump by those who qualify for the second jump. The distance of the second jump is independent of the distance of the first jump and is modelled with the same normal distribution. Students who jump a distance greater than 4.1 m in their second jump receive a certificate. At the start of the long jump competition, a student is selected at random.
4. Find the probability that this student will receive a certificate.
   

3. Hei and Tang are designing some pieces of art. They collected a large number of sticks. The random variable \(L\) represents the length of a stick in centimetres and has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). They sorted the sticks into lengths and painted them.
They found that \(60 \%\) of the sticks were longer than 45 cm and these were painted red, whilst \(15 \%\) of the sticks were shorter than 35 cm and these were painted blue. The remaining sticks were painted yellow.

1. Show that \(\mu\) and \(\sigma\) satisfy $$45 + 0.2533 \sigma = \mu$$
2. Find a second equation in \(\mu\) and \(\sigma\).
3. Hence find the value of \(\mu\) and the value of \(\sigma\).
4. Find
   1. \(\mathrm { P } ( L > 35 \mid L < 45 )\)
   2. \(\mathrm { P } ( L < 45 \mid L > 35 )\) Hei created her piece of art using a random selection of blue and yellow sticks.
      Tang created his piece of art using a random selection of red and yellow sticks.
      Hei and Tang each used the same number of sticks to create their piece of art.
      George is viewing Hei's and Tang's pieces of art. He finds a yellow stick on the floor that has fallen from one of these pieces.
5. With reference to your answers to part (d), state, giving a reason, whether the stick is more likely to have fallen from Hei's or Tang's piece of art.

Xiang is designing shelves for a bookshop. The height, \(H \mathrm {~cm}\), of books is modelled by the normal distribution with mean 25.1 cm and standard deviation 5.5 cm

1. Show that \(\mathrm { P } ( H > 30.8 ) = 0.15\) Xiang decided that the smallest \(5 \%\) of books and books taller than 30.8 cm would not be placed on the shelves. All the other books will be placed on the shelves.
2. Find the range of heights of books that will be placed on the shelves.
   (3) The books that will be placed on the shelves have heights classified as small, medium or large.
   The numbers of small, medium and large books are in the ratios \(2 : 3 : 3\)
3. The medium books have heights \(x \mathrm {~cm}\) where \(m < x < d\)
   1. Show that \(d = 25.8\) to 1 decimal place.
   2. Find the value of \(m\) Xiang wants 2 shelves for small books, 3 shelves for medium books and 3 shelves for large books.
      These shelves will be placed one above another and made of wood that is 1 cm thick.
4. Work out the minimum total height needed.

2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\).
   (4 marks)
   It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
   (2 marks)
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.
   (2 marks)

8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
4. State which company you would recommend. Give reasons for your answer.
   \begin{table}[h]
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   \end{table}

2. The lifetimes of batteries used for a computer game have a mean of 12 hours and a standard deviation of 3 hours. Battery lifetimes may be assumed to be normally distributed. Find the lifetime, \(t\) hours, of a battery such that 1 battery in 5 will have a lifetime longer than \(t\).

4 Chopped lettuce is sold in bags nominally containing 100 grams.
The weight, \(X\) grams, of chopped lettuce, delivered by the machine filling the bags, may be assumed to be normally distributed with mean \(\mu\) and standard deviation 4.

1. Assuming that \(\mu = 106\), determine the probability that a randomly selected bag of chopped lettuce:
   1. weighs less than 110 grams;
   2. is underweight.
2. Determine the minimum value of \(\mu\) so that at most 2 per cent of bags of chopped lettuce are underweight. Give your answer to one decimal place.
3. Boxes each contain 10 bags of chopped lettuce. The mean weight of a bag of chopped lettuce in a box is denoted by \(\bar { X }\). Given that \(\mu = 108.5\) :
   1. write down values for the mean and variance of \(\bar { X }\);
   2. determine the probability that \(\bar { X }\) exceeds 110 .

6 When Monica walks to work from home, she uses either route A or route B.

1. Her journey time, \(X\) minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8 . Determine:
   1. \(\mathrm { P } ( X < 45 )\);
   2. \(\mathrm { P } ( 30 < X < 45 )\).
2. Her journey time, \(Y\) minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of \(\sigma\). Given that \(\mathrm { P } ( Y > 45 ) = 0.12\), calculate the value of \(\sigma\).
3. If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am , state, with a reason, which route she should take.
4. When Monica travels to work from home by car, her journey time, \(W\) minutes, has a mean of 18 and a standard deviation of 12 . Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica's mean time is more than 20 minutes.
5. Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.

1 Draught excluder for doors and windows is sold in rolls of nominal length 10 metres.
The actual length, \(X\) metres, of draught excluder on a roll may be modelled by a normal distribution with mean 10.2 and standard deviation 0.15 .

1. Determine:
   1. \(\mathrm { P } ( X < 10.5 )\);
   2. \(\mathrm { P } ( 10.0 < X < 10.5 )\).
2. A customer randomly selects six 10 -metre rolls of the draught excluder. Calculate the probability that all six rolls selected contain more than 10 metres of draught excluder.

5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows. \(\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}\)

1. It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
   1. Construct a \(98 \%\) confidence interval for the mean weight, \(\mu\) grams, of flour in a bag, giving the limits to four significant figures.
   2. State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
   3. If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for \(\mu\) to be valid.
2. The statement ' 1 kg ' is printed on each bag. Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
3. Given that \(\mu = 1000\), state the probability that a \(98 \%\) confidence interval for \(\mu\) will not contain 1000.
   (l mark)

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]
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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.

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 \(100 \mathrm {~cm} ^ { 2 }\).