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

5 Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm .

1. \(8 \%\) of carrots are shorter than \(c \mathrm {~cm}\). Find the value of \(c\).
2. Rebekah picks 7 carrots at random. Find the probability that at least 2 of them have lengths between 15 and 16 cm .

1 It is given that \(X \sim \mathrm {~N} \left( 1.5,3.2 ^ { 2 } \right)\). Find the probability that a randomly chosen value of \(X\) is less than - 2.4 .

3 The amount of fibre in a packet of a certain brand of cereal is normally distributed with mean 160 grams. 19\% of packets of cereal contain more than 190 grams of fibre.

1. Find the standard deviation of the amount of fibre in a packet.
2. Kate buys 12 packets of cereal. Find the probability that at least 1 of the packets contains more than 190 grams of fibre.

5 On trains in the morning rush hour, each person is either a student with probability 0.36 , or an office worker with probability 0.22 , or a shop assistant with probability 0.29 or none of these.

1. 8 people on a morning rush hour train are chosen at random. Find the probability that between 4 and 6 inclusive are office workers.
2. 300 people on a morning rush hour train are chosen at random. Find the probability that between 31 and 49 inclusive are neither students nor office workers nor shop assistants.

2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .

5

1. The random variable \(X\) is normally distributed with mean 82 and standard deviation 7.4. Find the value of \(q\) such that \(\mathrm { P } ( 82 - q < X < 82 + q ) = 0.44\).
2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(5 \mu = 2 \sigma ^ { 2 }\) and that \(\mathrm { P } \left( Y < \frac { 1 } { 2 } \mu \right) = 0.281\). Find the values of \(\mu\) and \(\sigma\).

6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .

1. 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .
2. The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y \mathrm {~kg}\). Find the value of \(y\). Another farmer finds that the weights of sheep on his farm have a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .
3. Find the value of \(\mu\).

1 Packets of tea are labelled as containing 250 g . The actual weight of tea in a packet has a normal distribution with mean 260 g and standard deviation \(\sigma \mathrm { g }\). Any packet with a weight less than 250 g is classed as 'underweight'. Given that \(1 \%\) of packets of tea are underweight, find the value of \(\sigma\). [3]

5 Gem stones from a certain mine have weights, \(X\) grams, which are normally distributed with mean 1.9 g and standard deviation 0.55 g . These gem stones are sorted into three categories for sale depending on their weights, as follows. Small: under 1.2 g Medium: between 1.2 g and 2.5 g Large: over 2.5 g

1. Find the proportion of gem stones in each of these three categories.
2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 2.5 ) = 0.8\).

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 54.1 ) = 0.5\) and \(\mathrm { P } ( X > 50.9 ) = 0.8665\). Find the values of \(\mu\) and \(\sigma\).

4

1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.

7

1. A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
   1. Find on how many days of the year ( 365 days) the daily sales can be expected to exceed 3900 litres. The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
   2. Find the value of \(m\).
   3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a random value of \(Y\) is less than \(2 \mu\).

4 The time taken for cucumber seeds to germinate under certain conditions has a normal distribution with mean 125 hours and standard deviation \(\sigma\) hours.

1. It is found that \(13 \%\) of seeds take longer than 136 hours to germinate. Find the value of \(\sigma\).
2. 170 seeds are sown. Find the expected number of seeds which take between 131 and 141 hours to germinate.

1 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 20,49 )\). Given that \(\mathrm { P } ( X > k ) = 0.25\), find the value of \(k\).

4 Packets of rice are filled by a machine and have weights which are normally distributed with mean 1.04 kg and standard deviation 0.017 kg .

1. Find the probability that a randomly chosen packet weighs less than 1 kg .
2. How many packets of rice, on average, would the machine fill from 1000 kg of rice? The factory manager wants to produce more packets of rice. He changes the settings on the machine so that the standard deviation is the same but the mean is reduced to \(\mu \mathrm { kg }\). With this mean the probability that a packet weighs less than 1 kg is 0.0388 .
3. Find the value of \(\mu\).
4. How many packets of rice, on average, would the machine now fill from 1000 kg of rice?

4 The time taken to cook an egg by people living in a certain town has a normal distribution with mean 4.2 minutes and standard deviation 0.6 minutes.

1. Find the probability that a person chosen at random takes between 3.5 and 4.5 minutes to cook an egg. \(12 \%\) of people take more than \(t\) minutes to cook an egg.
2. Find the value of \(t\).
3. A random sample of \(n\) people is taken. Find the smallest possible value of \(n\) if the probability that none of these people takes more than \(t\) minutes to cook an egg is less than 0.003 .

6 The weights of bananas in a fruit shop have a normal distribution with mean 150 grams and standard deviation 50 grams. Three sizes of banana are sold. Small: under 95 grams
Medium: between 95 grams and 205 grams
Large: over 205 grams

1. Find the proportion of bananas that are small.
2. Find the weight exceeded by \(10 \%\) of bananas. The prices of bananas are 10 cents for a small banana, 20 cents for a medium banana and 25 cents for a large banana.
3. (a) Show that the probability that a randomly chosen banana costs 20 cents is 0.7286 .
   (b) Calculate the expected total cost of 100 randomly chosen bananas.

7 The weight, in grams, of pineapples is denoted by the random variable \(X\) which has a normal distribution with mean 500 and standard deviation 91.5. Pineapples weighing over 570 grams are classified as 'large'. Those weighing under 390 grams are classified as 'small' and the rest are classified as 'medium'.

1. Find the proportions of large, small and medium pineapples.
2. Find the weight exceeded by the heaviest \(5 \%\) of pineapples.
3. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 610 ) = 0.3\).

7 In Jimpuri the weights, in kilograms, of boys aged 16 years have a normal distribution with mean 61.4 and standard deviation 12.3.

1. Find the probability that a randomly chosen boy aged 16 years in Jimpuri weighs more than 65 kilograms.
2. For boys aged 16 years in Jimpuri, \(25 \%\) have a weight between 65 kilograms and \(k\) kilograms, where \(k\) is greater than 65 . Find \(k\).
   In Brigville the weights, in kilograms, of boys aged 16 years have a normal distribution. \(99 \%\) of the boys weigh less than 97.2 kilograms and \(33 \%\) of the boys weigh less than 55.2 kilograms.
3. Find the mean and standard deviation of the weights of boys aged 16 years in Brigville.

7 Josie aims to catch a bus which departs at a fixed time every day. Josie arrives at the bus stop \(T\) minutes before the bus departs, where \(T \sim \mathrm {~N} \left( 5.3,2.1 ^ { 2 } \right)\).

1. Find the probability that Josie has to wait longer than 6 minutes at the bus stop.
   On \(5 \%\) of days Josie has to wait longer than \(x\) minutes at the bus stop.
2. Find the value of \(x\).
3. Find the probability that Josie waits longer than \(x\) minutes on fewer than 3 days in 10 days.
4. Find the probability that Josie misses the bus.

4

1. It is given that \(X \sim \mathrm {~N} ( 31.4,3.6 )\). Find the probability that a randomly chosen value of \(X\) is less than 29.4.
2. The lengths of fish of a particular species are modelled by a normal distribution. A scientist measures the lengths of 400 randomly chosen fish of this species. He finds that 42 fish are less than 12 cm long and 58 are more than 19 cm long. Find estimates for the mean and standard deviation of the lengths of fish of this species.

7

1. The time, \(X\) hours, for which students use a games machine in any given day has a normal distribution with mean 3.24 hours and standard deviation 0.96 hours.
   1. On how many days of the year ( 365 days) would you expect a randomly chosen student to use a games machine for less than 4 hours?
   2. Find the value of \(k\) such that \(\mathrm { P } ( X > k ) = 0.2\).
   3. Find the probability that the number of hours for which a randomly chosen student uses a games machine in a day is within 1.5 standard deviations of the mean.
2. The variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), where \(4 \sigma = 3 \mu\) and \(\mu \neq 0\). Find the probability that a randomly chosen value of \(Y\) is positive.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The weights of apples sold by a store can be modelled by a normal distribution with mean 120 grams and standard deviation 24 grams. Apples weighing less than 90 grams are graded as 'small'; apples weighing more than 140 grams are graded as 'large'; the remainder are graded as 'medium'.

1. Show that the probability that an apple chosen at random is graded as medium is 0.692 , correct to 3 significant figures.
2. Four apples are chosen at random. Find the probability that at least two are graded as medium. [4]

6 The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation \(\sigma\) hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96 .

1. Find the value of \(\sigma\).
   New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.
2. For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.
3. Justify the use of your approximate distribution in part (ii).

7 The shortest time recorded by an athlete in a 400 m race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean 49.2 seconds and standard deviation 2.8 seconds.

1. Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds.
2. It is found that \(92 \%\) of athletes from this club have PBs of more than \(t\) seconds. Find the value of \(t\).
   Three athletes from the club are chosen at random.
3. Find the probability that exactly 2 have PBs of less than 46 seconds.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.