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

6 The heights, in metres, of fir trees in a large forest have a normal distribution with mean 40 and standard deviation 8 .

1. Find the probability that a fir tree chosen at random in this forest has a height less than 45 metres.
2. Find the probability that a fir tree chosen at random in this forest has a height within 5 metres of the mean.
   In another forest, the heights of another type of fir tree are modelled by a normal distribution. A scientist measures the heights of 500 randomly chosen trees of this type. He finds that 48 trees are less than 10 m high and 76 trees are more than 24 m high.
3. Find the mean and standard deviation of the heights of trees of this type.

4 The heights of students at the Mainland college are normally distributed with mean 148 cm and standard deviation 8 cm .

1. The probability that a Mainland student chosen at random has a height less than \(h \mathrm {~cm}\) is 0.67 . Find the value of \(h\).
   120 Mainland students are chosen at random.
2. Find the number of these students that would be expected to have a height within half a standard deviation of the mean.

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 \(\sigma\).

1 Name the distribution and suggest suitable numerical parameters that you could use to model the weights in kilograms of female 18-year-old students.

7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.

1. Find the standard deviation of the times spent by people visiting this dentist.
2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes.

5 The weights of letters posted by a certain business are normally distributed with mean 20 g . It is found that the weights of \(94 \%\) of the letters are within 12 g of the mean.

1. Find the standard deviation of the weights of the letters.
2. Find the probability that a randomly chosen letter weighs more than 13 g .
3. Find the probability that at least 2 of a random sample of 7 letters have weights which are more than 12 g above the mean.

6 There are a large number of students in Luttley College. \(60 \%\) of the students are boys. Students can choose exactly one of Games, Drama or Music on Friday afternoons. It is found that \(75 \%\) of the boys choose Games, \(10 \%\) of the boys choose Drama and the remainder of the boys choose Music. Of the girls, \(30 \%\) choose Games, \(55 \%\) choose Drama and the remainder choose Music.

1. 6 boys are chosen at random. Find the probability that fewer than 3 of them choose Music.
2. 5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.

1. In a certain country, the daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in winter has the distribution \(\mathrm { N } ( 8,24 )\). Find the probability that a randomly chosen winter day in this country has a minimum temperature between \(7 ^ { \circ } \mathrm { C }\) and \(12 ^ { \circ } \mathrm { C }\). The daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in another country in winter has a normal distribution with mean \(\mu\) and standard deviation \(2 \mu\).
2. Find the proportion of winter days on which the minimum temperature is below zero.
3. 70 winter days are chosen at random. Find how many of these would be expected to have a minimum temperature which is more than three times the mean.
4. The probability of the minimum temperature being above \(6 ^ { \circ } \mathrm { C }\) on any winter day is 0.0735 . Find the value of \(\mu\).

1 The random variable \(X\) is normally distributed and is such that the mean \(\mu\) is three times the standard deviation \(\sigma\). It is given that \(\mathrm { P } ( X < 25 ) = 0.648\).

1. Find the values of \(\mu\) and \(\sigma\).
2. Find the probability that, from 6 random values of \(X\), exactly 4 are greater than 25 .

3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .

1. Find the standard deviation of the lengths. Alison buys 8 rolls of parcel tape.
2. Find the probability that fewer than 3 of these rolls have lengths more than 77 m .

1

1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 25 )\).
   Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 30 )\).
2. A random variable \(Y\) has the distribution \(\mathrm { B } ( 100 , p )\) where \(p < 0.05\). Use the Poisson approximation to the binomial distribution to write down an expression, in terms of \(p\), for \(\mathrm { P } ( Y < 3 )\).

6 A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)\) and \(\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)\) respectively. Dried yeast costs \(\\) 13.50\( per kilogram and flour costs \)\\( 0.90\) per kilogram. For making one batch of bread the total of all other costs is \(\\) 55\(. The factory sells each batch of bread for \)\\( 200\). Find the probability that the profit made on one randomly chosen batch of bread is greater than \(\\) 40$. [7]

2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-04_2720_38_109_2010}

4 A population is normally distributed with mean 35 and standard deviation 8.1 . A random sample of size 140 is chosen from this population and the sample mean is denoted by \(\bar { X }\).

1. Find \(\mathrm { P } ( \bar { X } > 36 )\).
2. It is given that \(\mathrm { P } ( \bar { X } < a ) = 0.986\). Find the value of \(a\).

6 When Sunil travels from his home in England to visit his relatives in India, his journey is in four stages. The times, in hours, for the stages have independent normal distributions as follows. Bus from home to the airport: \(\quad \mathrm { N } ( 3.75,1.45 )\) Waiting in the airport: \(\quad \mathrm { N } ( 3.1,0.785 )\) Flight from England to India: \(\quad \mathrm { N } ( 11,1.3 )\) Car in India to relatives: \(\quad \mathrm { N } ( 3.2,0.81 )\)

1. Find the probability that the flight time is shorter than the total time for the other three stages.
2. Find the probability that, for 6 journeys to India, the mean time waiting in the airport is less than 4 hours.

3 Metal bolts are produced in large numbers and have lengths which are normally distributed with mean 2.62 cm and standard deviation 0.30 cm .

1. Find the probability that a random sample of 45 bolts will have a mean length of more than 2.55 cm .
2. The machine making these bolts is given an annual service. This may change the mean length of bolts produced but does not change the standard deviation. To test whether the mean has changed, a random sample of 30 bolts is taken and their lengths noted. The sample mean length is \(m \mathrm {~cm}\). Find the set of values of \(m\) which result in rejection at the \(10 \%\) significance level of the hypothesis that no change in the mean length has occurred.

4 The weekly distance in kilometres driven by Mr Parry has a normal distribution with mean 512 and standard deviation 62. Independently, the weekly distance in kilometres driven by Mrs Parry has a normal distribution with mean 89 and standard deviation 7.4.

1. Find the probability that, in a randomly chosen week, Mr Parry drives more than 5 times as far as Mrs Parry.
2. Find the mean and standard deviation of the total of the weekly distances in miles driven by Mr Parry and Mrs Parry. Use the approximation 8 kilometres \(= 5\) miles.

3 The weight, in grams, of a certain type of apple is modelled by the random variable \(X\) with mean 62 and standard deviation 8.2. A random sample of 50 apples is selected, and the mean weight in grams, \(\bar { X }\), is found.

1. Describe fully the distribution of \(\bar { X }\).
2. Find \(\mathrm { P } ( \bar { X } > 64 )\).

6 Yu Ming travels to work and returns home once each day. The times, in minutes, that he takes to travel to work and to return home are represented by the independent random variables \(W\) and \(H\) with distributions \(\mathrm { N } \left( 22.4,4.8 ^ { 2 } \right)\) and \(\mathrm { N } \left( 20.3,5.2 ^ { 2 } \right)\) respectively.

1. Find the probability that Yu Ming's total travelling time during a 5-day period is greater than 180 minutes.
2. Find the probability that, on a particular day, Yu Ming takes longer to return home than he takes to travel to work.

5 Each drink from a coffee machine contains \(X \mathrm {~cm} ^ { 3 }\) of coffee and \(Y \mathrm {~cm} ^ { 3 }\) of milk, where \(X\) and \(Y\) are independent variables with \(X \sim \mathrm {~N} \left( 184,15 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 50,8 ^ { 2 } \right)\). If the total volume of the drink is less than \(200 \mathrm {~cm} ^ { 3 }\) the customer receives the drink without charge.

1. Find the percentage of drinks which customers receive without charge.
2. Find the probability that, in a randomly chosen drink, the volume of coffee is more than 4 times the volume of milk.

7 Previous records have shown that the number of cars entering Bampor on any day has mean 352 and variance 121.

1. Find the probability that the mean number of cars entering Bampor during a random sample of 200 days is more than 354 .
2. State, with a reason, whether it was necessary to assume that the number of cars entering Bampor on any day has a normal distribution in order to find the probability in part (i).
3. It is thought that the population mean may recently have changed. The number of cars entering Bampor during the day was recorded for each of a random sample of 50 days and the sample mean was found to be 356 . Assuming that the variance is unchanged, test at the \(5 \%\) significance level whether the population mean is still 352 .

5 Packets of cereal are packed in boxes, each containing 6 packets. The masses of the packets are normally distributed with mean 510 g and standard deviation 12 g . The masses of the empty boxes are normally distributed with mean 70 g and standard deviation 4 g .

1. Find the probability that the total mass of a full box containing 6 packets is between 3050 g and 3150 g .
2. A packet and an empty box are chosen at random. Find the probability that the mass of the packet is at least 8 times the mass of the empty box.

1 The masses, in grams, of apples of a certain type are normally distributed with mean 60.4 and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than 436 g is rejected. Find the proportion of bags that are rejected.

4 The proportion of people who have a particular gene, on average, is 1 in 1000. A random sample of 3500 people in a certain country is chosen and the number of people, \(X\), having the gene is found.

1. State the distribution of \(X\) and state also an appropriate approximating distribution. Give the values of any parameters in each case. Justify your choice of the approximating distribution.
2. Use the approximating distribution to find \(\mathrm { P } ( X \leqslant 3 )\).

8 In an examination, the marks in the theory paper and the marks in the practical paper are denoted by the random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 57,13 )\) and \(Y \sim \mathrm {~N} ( 28,5 )\). You may assume that each candidate's marks in the two papers are independent. The final score of each candidate is found by calculating \(X + 2.5 Y\). A candidate is chosen at random. Without using a continuity correction, find the probability that this candidate

1. has a final score that is greater than 140 ,
2. obtains at least 20 more marks in the theory paper than in the practical paper.

4 The masses, in grams, of large bags of sugar and small bags of sugar are denoted by \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 5.1,0.2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 2.5,0.1 ^ { 2 } \right)\). Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag.