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

4 Trees in the Redian forest are classified as tall, medium or short, according to their height. The heights can be modelled by a normal distribution with mean 40 m and standard deviation 12 m . Trees with a height of less than 25 m are classified as short.

1. Find the probability that a randomly chosen tree is classified as short.
   Of the trees that are classified as tall or medium, one third are tall and two thirds are medium.
2. Show that the probability that a randomly chosen tree is classified as tall is 0.298 , correct to 3 decimal places.
3. Find the height above which trees are classified as tall.

3 In a certain town, the time, \(X\) hours, for which people watch television in a week has a normal distribution with mean 15.8 hours and standard deviation 4.2 hours.

1. Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week.
2. Find the value of \(k\) such that \(\mathrm { P } ( X < k ) = 0.75\).

2 A company produces a particular type of metal rod. The lengths of these rods are normally distributed with mean 25.2 cm and standard deviation 0.4 cm . A random sample of 500 of these rods is chosen. How many rods in this sample would you expect to have a length that is within 0.5 cm of the mean length?

2 The weights of bags of sugar are normally distributed with mean 1.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 2000 bags of sugar, 72 weighed more than 1.10 kg . Find the value of \(\sigma\).

5 The lengths of the leaves of a particular type of tree are modelled by a normal distribution. A scientist measures the lengths of a random sample of 500 leaves from this type of tree and finds that 42 are less than 4 cm long and 100 are more than 10 cm long.

1. Find estimates for the mean and standard deviation of the lengths of leaves from this type of tree.
   The lengths, in cm , of the leaves of a different type of tree have the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The scientist takes a random sample of 800 leaves from this type of tree.
2. Find how many of these leaves the scientist would expect to have lengths, in cm , between \(\mu - 2 \sigma\) and \(\mu + 2 \sigma\).

5 The lengths, in cm, of the leaves of a particular type are modelled by the distribution \(\mathrm { N } \left( 5.2,1.5 ^ { 2 } \right)\).

1. Find the probability that a randomly chosen leaf of this type has length less than 6 cm .
   The lengths of the leaves of another type are also modelled by a normal distribution. A scientist measures the lengths of a random sample of 500 leaves of this type and finds that 46 are less than 3 cm long and 95 are more than 8 cm long.
2. Find estimates for the mean and standard deviation of the lengths of leaves of this type.
3. In a random sample of 2000 leaves of this second type, how many would the scientist expect to find with lengths more than 1 standard deviation from the mean?

4 Ramesh throws an ordinary fair 6-sided die.

1. Find the probability that he obtains a 4 for the first time on his 8th throw.
2. Find the probability that it takes no more than 5 throws for Ramesh to obtain a 4 .
   Ramesh now repeatedly throws two ordinary fair 6-sided dice at the same time. Each time he adds the two numbers that he obtains.
3. For 10 randomly chosen throws of the two dice, find the probability that Ramesh obtains a total of less than 4 on at least three throws.

5 Farmer Jones grows apples. The weights, in grams, of the apples grown this year are normally distributed with mean 170 and standard deviation 25. Apples that weigh between 142 grams and 205 grams are sold to a supermarket.

1. Find the probability that a randomly chosen apple grown by Farmer Jones this year is sold to the supermarket.
   Farmer Jones sells the apples to the supermarket at \(\\) 0.24\( each. He sells apples that weigh more than 205 grams to a local shop at \)\\( 0.30\) each. He does not sell apples that weigh less than 142 grams. The total number of apples grown by Farmer Jones this year is 20000.
2. Calculate an estimate for his total income from this year's apples.
   Farmer Tan also grows apples. The weights, in grams, of the apples grown this year follow the distribution \(\mathrm { N } \left( 182,20 ^ { 2 } \right) .72 \%\) of these apples have a weight more than \(w\) grams.
3. Find the value of \(w\).

4 A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minutes and standard deviation 5.2 minutes.

1. In a random sample of 250 of the students, how many would you expect to have taken more than 20 minutes to complete the puzzle?
   All the students are given a second puzzle to complete. Their times, in minutes, are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(20 \%\) of the students have times less than 14.5 minutes and \(67 \%\) of the students have times greater than 18.5 minutes.
2. Find the value of \(\mu\) and the value of \(\sigma\).

5 The lengths of Western bluebirds are normally distributed with mean 16.5 cm and standard deviation 0.6 cm . A random sample of 150 of these birds is selected.

1. How many of these 150 birds would you expect to have length between 15.4 cm and 16.8 cm ?
   The lengths of Eastern bluebirds are normally distributed with mean 18.4 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that \(72 \%\) of Eastern bluebirds have length greater than 17.1 cm .
2. Find the value of \(\sigma\).
   A random sample of 120 Eastern bluebirds is chosen.
3. Use an approximation to find the probability that fewer than 80 of these 120 bluebirds have length greater than 17.1 cm .

6 The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On \(10 \%\) of days less than 16 kg of grapes are sold.

1. Find the standard deviation of the mass of grapes sold per day.
   The mass of grapes sold on any day is independent of the mass sold on any other day.
2. 12 days are chosen at random. Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.
3. In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?

2 The lengths of the tails of adult raccoons of a certain species are normally distributed with mean 28 cm and standard deviation 3.3 cm .

1. Find the probability that a randomly chosen adult raccoon of this species has a tail length between 23 cm and 35 cm .
   The masses of adult raccoons of this species are normally distributed with mean 8.5 kg and standard deviation \(\sigma \mathrm { kg } .75 \%\) of adult raccoons of this species have mass greater than 7.6 kg .
2. Find the value of \(\sigma\).

3 The weights of oranges can be modelled by a normal distribution with mean 131 grams and standard deviation 54 grams. Oranges are classified as small, medium or large. A large orange weighs at least 184 grams and 20\% of oranges are classified as small.

1. Find the percentage of oranges that are classified as large.
2. Find the greatest possible weight of a small orange.

2 In a certain country, the heights of the adult population are normally distributed with mean 1.64 m and standard deviation 0.25 m .

1. Find the probability that an adult chosen at random from this country will have height greater than 1.93 m . \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-04_2716_35_143_2012} \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-05_2724_35_136_20} In another country, the heights of the adult population are also normally distributed. \(33 \%\) of the adult population have height less than \(1.56 \mathrm {~m} .25 \%\) of the adult population have height greater than 1.86 m .
2. Find the mean and the standard deviation of this distribution.

3 The weights of apples of a certain variety are normally distributed with mean 82 grams. \(22 \%\) of these apples have a weight greater than 87 grams.

1. Find the standard deviation of the weights of these apples.
2. Find the probability that the weight of a randomly chosen apple of this variety differs from the mean weight by less than 4 grams.

3 The time spent by shoppers in a large shopping centre has a normal distribution with mean 96 minutes and standard deviation 18 minutes.

1. Find the probability that a shopper chosen at random spends between 85 and 100 minutes in the shopping centre. \(88 \%\) of shoppers spend more than \(t\) minutes in the shopping centre.
2. Find the value of \(t\).

4 The weights of male leopards in a particular region are normally distributed with mean 55 kg and standard deviation 6 kg .

1. Find the probability that a randomly chosen male leopard from this region weighs between 46 and 62 kg .
   The weights of female leopards in this region are normally distributed with mean 42 kg and standard deviation \(\sigma \mathrm { kg }\). It is known that \(25 \%\) of female leopards in the region weigh less than 36 kg .
2. Find the value of \(\sigma\).
   The distributions of the weights of male and female leopards are independent of each other. A male leopard and a female leopard are each chosen at random.
3. Find the probability that both the weights of these leopards are less than 46 kg .

4 A company sells small and large bags of rice. The masses of the small bags of rice are normally distributed with mean 1.20 kg and standard deviation 0.16 kg .

1. In a random sample of 500 of these small bags of rice, how many would you expect to have a mass greater than 1.26 kg ?
   The masses of the large bags of rice are normally distributed with mean 2.50 kg and standard deviation \(\sigma \mathrm { kg } .20 \%\) of these large bags of rice have a mass less than 2.40 kg .
2. Find the value of \(\sigma\).
   A random sample of 80 large bags of rice is chosen.
3. Use a suitable approximation to find the probability that fewer than 22 of these large bags of rice have a mass less than 2.40 kg .

5 The time in hours that Davin plays on his games machine each day is normally distributed with mean 3.5 and standard deviation 0.9.

1. Find the probability that on a randomly chosen day Davin plays on his games machine for more than 4.2 hours.
2. On 90\% of days Davin plays on his games machine for more than \(t\) hours. Find the value of \(t\).
3. Calculate an estimate for the number of days in a year ( 365 days) on which Davin plays on his games machine for between 2.8 and 4.2 hours.

3 Pia runs 2 km every day and her times in minutes are normally distributed with mean 10.1 and standard deviation 1.3.

1. Find the probability that on a randomly chosen day Pia takes longer than 11.3 minutes to run 2 km .
2. On \(75 \%\) of days, Pia takes longer than \(t\) minutes to run 2 km . Find the value of \(t\).
3. On how many days in a period of 90 days would you expect Pia to take between 8.9 and 11.3 minutes to run 2 km ?

1 The times taken to swim 100 metres by members of a large swimming club have a normal distribution with mean 62 seconds and standard deviation 5 seconds.

1. Find the probability that a randomly chosen member of the club takes between 56 and 66 seconds to swim 100 metres.
2. \(13 \%\) of the members of the club take more than \(t\) minutes to swim 100 metres. Find the value of \(t\).

7 The times, in minutes, that Karli spends each day on social media are normally distributed with mean 125 and standard deviation 24.

1. 1. On how many days of the year ( 365 days) would you expect Karli to spend more than 142 minutes on social media?
   2. Find the probability that Karli spends more than 142 minutes on social media on fewer than 2 of 10 randomly chosen days.
2. On \(90 \%\) of days, Karli spends more than \(t\) minutes on social media. Find the value of \(t\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 Raj wants to improve his fitness, so every day he goes for a run. The times, in minutes, of his runs have a normal distribution with mean 41.2 and standard deviation 3.6.

1. Find the probability that on a randomly chosen day Raj runs for more than 43.2 minutes.
2. Find an estimate for the number of days in a year ( 365 days) on which Raj runs for less than 43.2 minutes.
3. On 95\% of days, Raj runs for more than \(t\) minutes. Find the value of \(t\).

4 In a large population, the systolic blood pressure (SBP) of adults is normally distributed with mean 125.4 and standard deviation 18.6.

1. Find the probability that the SBP of a randomly chosen adult is less than 132.
   The SBP of 12-year-old children in the same population is normally distributed with mean 117. Of these children 88\% have SBP more than 108.
2. Find the standard deviation of this distribution.
   Three adults are chosen at random from this population.
3. Find the probability that each of these three adults has SBP within 1.5 standard deviations of the mean.

2 The lengths of the rods produced by a company are normally distributed with mean 55.6 mm and standard deviation 1.2 mm .

1. In a random sample of 400 of these rods, how many would you expect to have length less than 54.8 mm ?
2. Find the probability that a randomly chosen rod produced by this company has a length that is within half a standard deviation of the mean.