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

2 In a large college, \(32 \%\) of the students have blue eyes. A random sample of 80 students is chosen. Use an approximation to find the probability that fewer than 20 of these students have blue eyes.

5 Company \(A\) produces bags of sugar. An inspector finds that on average \(10 \%\) of the bags are underweight. 10 of the bags are chosen at random.

1. Find the probability that fewer than 3 of these bags are underweight.
   The weights of the bags of sugar produced by company \(B\) are normally distributed with mean 1.04 kg and standard deviation 0.06 kg .
2. Find the probability that a randomly chosen bag produced by company \(B\) weighs more than 1.11 kg . \(81 \%\) of the bags of sugar produced by company \(B\) weigh less than \(w \mathrm {~kg}\).
3. Find the value of \(w\).

3 A farmer sells eggs. The weights, in grams, of the eggs can be modelled by a normal distribution with mean 80.5 and standard deviation 6.6. Eggs are classified as small, medium or large according to their weight. A small egg weighs less than 76 grams and \(40 \%\) of the eggs are classified as medium.

1. Find the percentage of eggs that are classified as small.
2. Find the least possible weight of an egg classified as large.
   150 of the eggs for sale last week were weighed.
3. Use an approximation to find the probability that more than 68 of these eggs were classified as medium.

3 A factory produces a certain type of electrical component. It is known that \(15 \%\) of the components produced are faulty. A random sample of 200 components is chosen. Use an approximation to find the probability that more than 40 of these components are faulty.

5

1. The heights of the members of a club are normally distributed with mean 166 cm and standard deviation 10 cm .
   1. Find the probability that a randomly chosen member of the club has height less than 170 cm .
   2. Given that \(40 \%\) of the members have heights greater than \(h \mathrm {~cm}\), find the value of \(h\) correct to 2 decimal places.
2. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a randomly chosen value of \(X\) is positive.

2 The weights of large bags of pasta produced by a company are normally distributed with mean 1.5 kg and standard deviation 0.05 kg .

1. Find the probability that a randomly chosen large bag of pasta weighs between 1.42 kg and 1.52 kg .
   The weights of small bags of pasta produced by the company are normally distributed with mean 0.75 kg and standard deviation \(\sigma \mathrm { kg }\). It is found that \(68 \%\) of these small bags have weight less than 0.9 kg .
2. Find the value of \(\sigma\).

5 The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams.

1. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-09_2717_29_105_22}
2. The shop also sells red apples. \(60 \%\) of the red apples sold by the shop weigh more than 80 grams. 160 red apples are chosen at random from the shop. Use a suitable approximation to find the probability that fewer than 105 of the chosen red apples weigh more than 80 grams.

6 The heights of the female students at Breven college are normally distributed:

• \(90 \%\) of the female students have heights less than 182.7 cm .
• \(40 \%\) of the female students have heights less than 162.5 cm .
  1. Find the mean and the standard deviation of the heights of the female students at Breven college. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-10_2715_41_110_2008} \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-11_2723_35_101_20}

Ten female students are chosen at random from those at Breven college.

Find the probability that fewer than 8 of these 10 students have heights more than 162.5 cm .

4 The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees.

1. How many of these trees would you expect to have height less than 18.2 metres?
   The heights, in metres, of red pine trees are normally distributed with mean 23.4 and standard deviation \(\sigma\). It is known that \(26 \%\) of red pine trees have height greater than 25.5 metres.
2. Find the value of \(\sigma\).

3 In Molimba, the heights, in cm , of adult males are normally distributed with mean 176 cm and standard deviation 4.8 cm .

1. Find the probability that a randomly chosen adult male in Molimba has a height greater than 170 cm .
   60\% of adult males in Molimba have a height between 170 cm and \(k \mathrm {~cm}\), where \(k\) is greater than 170 .
2. Find the value of \(k\), giving your answer correct to 1 decimal place.

3

1. The height of sunflowers follows a normal distribution with mean 112 cm and standard deviation 17.2 cm . Find the probability that the height of a randomly chosen sunflower is greater than 120 cm .
2. When a new fertiliser is used, the height of sunflowers follows a normal distribution with mean 115 cm . Given that \(80 \%\) of the heights are now greater than 103 cm , find the standard deviation.

6 The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm .

1. Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm .
   The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than 45 cm and 17 have lengths more than 56 cm .
2. Find estimates for the mean and standard deviation of the lengths of male snakes of this species.

4 Melons are sold in three sizes: small, medium and large. The weights follow a normal distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than 350 grams are classified as small.

1. Find the proportion of melons which are classified as small.
2. The rest of the melons are divided in equal proportions between medium and large. Find the weight above which melons are classified as large.

6 Tyre pressures on a certain type of car independently follow a normal distribution with mean 1.9 bars and standard deviation 0.15 bars.

1. Find the probability that all four tyres on a car of this type have pressures between 1.82 bars and 1.92 bars.
2. Safety regulations state that the pressures must be between \(1.9 - b\) bars and \(1.9 + b\) bars. It is known that \(80 \%\) of tyres are within these safety limits. Find the safety limits.

3 The lengths of fish of a certain type have a normal distribution with mean 38 cm . It is found that \(5 \%\) of the fish are longer than 50 cm .

1. Find the standard deviation.
2. When fish are chosen for sale, those shorter than 30 cm are rejected. Find the proportion of fish rejected.
3. 9 fish are chosen at random. Find the probability that at least one of them is longer than 50 cm .

3

1. The random variable \(X\) is normally distributed. The mean is twice the standard deviation. It is given that \(\mathrm { P } ( X > 5.2 ) = 0.9\). Find the standard deviation.
2. A normal distribution has mean \(\mu\) and standard deviation \(\sigma\). If 800 observations are taken from this distribution, how many would you expect to be between \(\mu - \sigma\) and \(\mu + \sigma\) ?

4 In a certain country the time taken for a common infection to clear up is normally distributed with mean \(\mu\) days and standard deviation 2.6 days. \(25 \%\) of these infections clear up in less than 7 days.

1. Find the value of \(\mu\). In another country the standard deviation of the time taken for the infection to clear up is the same as in part (i), but the mean is 6.5 days. The time taken is normally distributed.
2. Find the probability that, in a randomly chosen case from this country, the infection takes longer than 6.2 days to clear up.

1 The volume of milk in millilitres in cartons is normally distributed with mean \(\mu\) and standard deviation 8. Measurements were taken of the volume in 900 of these cartons and it was found that 225 of them contained more than 1002 millilitres.

1. Calculate the value of \(\mu\).
2. Three of these 900 cartons are chosen at random. Calculate the probability that exactly 2 of them contain more than 1002 millilitres.

5

1. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(3 \mu = 7 \sigma ^ { 2 }\) and that \(\mathrm { P } ( X > 2 \mu ) = 0.1016\). Find \(\mu\) and \(\sigma\).
2. It is given that \(Y \sim \mathrm {~N} ( 33,21 )\). Find the value of \(a\) given that \(\mathrm { P } ( 33 - a < Y < 33 + a ) = 0.5\).

6 The lengths, in centimetres, of drinking straws produced in a factory have a normal distribution with mean \(\mu\) and variance 0.64 . It is given that \(10 \%\) of the straws are shorter than 20 cm .

1. Find the value of \(\mu\).
2. Find the probability that, of 4 straws chosen at random, fewer than 2 will have a length between 21.5 cm and 22.5 cm .

5 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\frac { 1 } { 4 } \mu\). It is given that \(\mathrm { P } ( X > 20 ) = 0.04\).

1. Find \(\mu\).
2. Find \(\mathrm { P } ( 10 < X < 20 )\).
3. 250 independent observations of \(X\) are taken. Find the probability that at least 235 of them are less than 20.

1 It is given that \(X \sim \mathrm {~N} ( 28.3,4.5 )\). Find the probability that a randomly chosen value of \(X\) lies between 25 and 30 .

6 The lengths of body feathers of a particular species of bird are modelled by a normal distribution. A researcher measures the lengths of a random sample of 600 body feathers from birds of this species and finds that 63 are less than 6 cm long and 155 are more than 12 cm long.

1. Find estimates of the mean and standard deviation of the lengths of body feathers of birds of this species.
2. In a random sample of 1000 body feathers from birds of this species, how many would the researcher expect to find with lengths more than 1 standard deviation from the mean?

7 The times taken to play Beethoven's Sixth Symphony can be assumed to have a normal distribution with mean 41.1 minutes and standard deviation 3.4 minutes. Three occasions on which this symphony is played are chosen at random.

1. Find the probability that the symphony takes longer than 42 minutes to play on exactly 1 of these occasions. The times taken to play Beethoven's Fifth Symphony can also be assumed to have a normal distribution. The probability that the time is less than 26.5 minutes is 0.1 , and the probability that the time is more than 34.6 minutes is 0.05 .
2. Find the mean and standard deviation of the times to play this symphony.
3. Assuming that the times to play the two symphonies are independent of each other, find the probability that, when both symphonies are played, both of the times are less than 34.6 minutes.

6 The lengths, in cm, of trout in a fish farm are normally distributed. 96\% of the lengths are less than 34.1 cm and 70\% of the lengths are more than 26.7 cm .

1. Find the mean and the standard deviation of the lengths of the trout. In another fish farm, the lengths of salmon, \(X \mathrm {~cm}\), are normally distributed with mean 32.9 cm and standard deviation 2.4 cm .
2. Find the probability that a randomly chosen salmon is 34 cm long, correct to the nearest centimetre.
3. Find the value of \(t\) such that \(\mathrm { P } ( 31.8 < X < t ) = 0.5\).