2.04f Find normal probabilities: Z transformation

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1. The random variable \(Y\) is normally distributed with positive mean \(\mu\) and standard deviation \(\frac { 1 } { 2 } \mu\). Find the probability that a randomly chosen value of \(Y\) is negative.
2. The weights of bags of rice are normally distributed with mean 2.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 8000 such bags, 253 weighed over 2.1 kg . Find the value of \(\sigma\). [4]

1 The random variable \(Y\) is normally distributed with mean equal to five times the standard deviation. It is given that \(\mathrm { P } ( Y > 20 ) = 0.0732\). Find the mean.

3 Cans of lemon juice are supposed to contain 440 ml of juice. It is found that the actual volume of juice in a can is normally distributed with mean 445 ml and standard deviation 3.6 ml .

1. Find the probability that a randomly chosen can contains less than 440 ml of juice. It is found that \(94 \%\) of the cans contain between \(( 445 - c ) \mathrm { ml }\) and \(( 445 + c ) \mathrm { ml }\) of juice.
2. Find the value of \(c\).

3 Buildings in a certain city centre are classified by height as tall, medium or short. The heights can be modelled by a normal distribution with mean 50 metres and standard deviation 16 metres. Buildings with a height of more than 70 metres are classified as tall.

1. Find the probability that a building chosen at random is classified as tall.
2. The rest of the buildings are classified as medium and short in such a way that there are twice as many medium buildings as there are short ones. Find the height below which buildings are classified as short.

1 The petrol consumption of a certain type of car has a normal distribution with mean 24 kilometres per litre and standard deviation 4.7 kilometres per litre. Find the probability that the petrol consumption of a randomly chosen car of this type is between 21.6 kilometres per litre and 28.7 kilometres per litre.

2 Lengths of a certain type of white radish are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm } .4 \%\) of these radishes are longer than 12 cm and \(32 \%\) are longer than 9 cm . Find \(\mu\) and \(\sigma\).

7 The time Rafa spends on his homework each day in term-time has a normal distribution with mean 1.9 hours and standard deviation \(\sigma\) hours. On \(80 \%\) of these days he spends more than 1.35 hours on his homework.

1. Find the value of \(\sigma\).
2. Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework.
3. A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than 1.35 hours on his homework is between 163 and 173 inclusive.

5 When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.

1. \(90 \%\) of Moses's phone calls take longer than \(t\) minutes. Find the value of \(t\).
2. Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean.

1 The lengths, in metres, of cars in a city are normally distributed with mean \(\mu\) and standard deviation 0.714 . The probability that a randomly chosen car has a length more than 3.2 metres and less than \(\mu\) metres is 0.475 . Find \(\mu\).

1 The weights, in grams, of onions in a supermarket have a normal distribution with mean \(\mu\) and standard deviation 22. The probability that a randomly chosen onion weighs more than 195 grams is 0.128 . Find the value of \(\mu\).

5 The heights of books in a library, in cm, have a normal distribution with mean 21.7 and standard deviation 6.5. A book with a height of more than 29 cm is classified as 'large'.

1. Find the probability that, of 8 books chosen at random, fewer than 2 books are classified as large.
2. \(n\) books are chosen at random. The probability of there being at least 1 large book is more than 0.98 . Find the least possible value of \(n\).

1 The height of maize plants in Mpapwa is normally distributed with mean 1.62 m and standard deviation \(\sigma \mathrm { m }\). The probability that a randomly chosen plant has a height greater than 1.8 m is 0.15 . Find the value of \(\sigma\).

5 Plastic drinking straws are manufactured to fit into drinks cartons which have a hole in the top. A straw fits into the hole if the diameter of the straw is less than 3 mm . The diameters of the straws have a normal distribution with mean 2.6 mm and standard deviation 0.25 mm .

1. A straw is chosen at random. Find the probability that it fits into the hole in a drinks carton.
2. 500 straws are chosen at random. Use a suitable approximation to find the probability that at least 480 straws fit into the holes in drinks cartons.
3. Justify the use of your approximation.

2 When visiting the dentist the probability of waiting less than 5 minutes is 0.16 , and the probability of waiting less than 10 minutes is 0.88 .

1. Find the probability of waiting between 5 and 10 minutes. A random sample of 180 people who visit the dentist is chosen.
2. Use a suitable approximation to find the probability that more than 115 of these people wait between 5 and 10 minutes.

6 The time in minutes taken by Peter to walk to the shop and buy a newspaper is normally distributed with mean 9.5 and standard deviation 1.3.

1. Find the probability that on a randomly chosen day Peter takes longer than 10.2 minutes.
2. On \(90 \%\) of days he takes longer than \(t\) minutes. Find the value of \(t\).
3. Calculate an estimate of the number of days in a year ( 365 days) on which Peter takes less than 8.8 minutes to walk to the shop and buy a newspaper.

5 The heights of school desks have a normal distribution with mean 69 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that 15.5\% of these desks have a height greater than 70 cm .

1. Find the value of \(\sigma\). When Jodu sits at a desk, his knees are at a height of 58 cm above the floor. A desk is comfortable for Jodu if his knees are at least 9 cm below the top of the desk. Jodu's school has 300 desks.
2. Calculate an estimate of the number of these desks that are comfortable for Jodu.

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1. The random variable \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). You are given that \(\sigma = 0.25 \mu\) and \(\mathrm { P } ( X < 6.8 ) = 0.75\).
   1. Find the value of \(\mu\).
   2. Find \(\mathrm { P } ( X < 4.7 )\).
2. The lengths of metal rods have a normal distribution with mean 16 cm and standard deviation 0.2 cm . Rods which are shorter than 15.75 cm or longer than 16.25 cm are not usable. Find the expected number of usable rods in a batch of 1000 rods.

5 The lengths of videos of a certain popular song have a normal distribution with mean 3.9 minutes. \(18 \%\) of these videos last for longer than 4.2 minutes.

1. Find the standard deviation of the lengths of these videos.
2. Find the probability that the length of a randomly chosen video differs from the mean by less than half a minute.
   The lengths of videos of another popular song have a normal distribution with the same mean of 3.9 minutes but the standard deviation is twice the standard deviation in part (i). The probability that the length of a randomly chosen video of this song differs from the mean by less than half a minute is denoted by \(p\).
3. Without any further calculation, determine whether \(p\) is more than, equal to, or less than your answer to part (ii). You must explain your reasoning.

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1. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu = 1.5 \sigma\). A random value of \(X\) is chosen. Find the probability that this value of \(X\) is greater than 0 .
2. The life of a particular type of torch battery is normally distributed with mean 120 hours and standard deviation \(s\) hours. It is known that \(87.5 \%\) of these batteries last longer than 70 hours. Find the value of \(s\).

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1. The distance that car tyres of a certain make can travel before they need to be replaced has a normal distribution. A survey of a large number of these tyres found that the probability of this distance being more than 36800 km is 0.0082 and the probability of this distance being more than 31000 km is 0.6915 . Find the mean and standard deviation of the distribution.
2. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(3 \sigma = 4 \mu\) and \(\mu \neq 0\). Find \(\mathrm { P } ( X < 3 \mu )\). [3]

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1. The volume of soup in Super Soup cartons has a normal distribution with mean \(\mu\) millilitres and standard deviation 9 millilitres. Tests have shown that \(10 \%\) of cartons contain less than 440 millilitres of soup. Find the value of \(\mu\).
2. A food retailer orders 150 Super Soup cartons. Calculate the number of these cartons for which you would expect the volume of soup to be more than 1.8 standard deviations above the mean.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( - 3 , \sigma ^ { 2 } \right)\). The probability that a randomly chosen value of \(X\) is positive is 0.25 .

1. Find the value of \(\sigma\).
2. Find the probability that, of 8 random values of \(X\), fewer than 2 will be positive.

6 The diameters of apples in an orchard have a normal distribution with mean 5.7 cm and standard deviation 0.8 cm . Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.

1. Find the probability that an apple selected at random can be used as a toffee apple.
2. 250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.

7 The weight of adult female giraffes has a normal distribution with mean 830 kg and standard deviation 120 kg .

1. There are 430 adult female giraffes in a particular game reserve. Find the number of these adult female giraffes which can be expected to weigh less than 700 kg .
2. Given that \(90 \%\) of adult female giraffes weigh between \(( 830 - w ) \mathrm { kg }\) and \(( 830 + w ) \mathrm { kg }\), find the value of \(w\).
   The weight of adult male giraffes has a normal distribution with mean 1190 kg and standard deviation \(\sigma \mathrm { kg }\).
3. Given that \(83.4 \%\) of adult male giraffes weigh more than 950 kg , find the value of \(\sigma\).

2 The volume of ink in a certain type of ink cartridge has a normal distribution with mean 30 ml and standard deviation 1.5 ml . People in an office use a total of 8 cartridges of this ink per month. Find the expected number of cartridges per month that contain less than 28.9 ml of this ink.