2.04f Find normal probabilities: Z transformation

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.

5 The score on one throw of a 4 -sided die is denoted by the random variable \(X\) with probability distribution as shown in the table.

1. Show that \(\operatorname { Var } ( X ) = 1.25\). The die is thrown 300 times. The score on each throw is noted and the mean, \(\bar { X }\), of the 300 scores is found.
2. Use a normal distribution to find \(\mathrm { P } ( \bar { X } < 1.4 )\).
3. Justify the use of the normal distribution in part (ii).

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.

7 The volume of liquid in cans of cola is normally distributed with mean 330 millilitres and standard deviation 5.2 millilitres. The volume of liquid in bottles of tonic water is normally distributed with mean 500 millilitres and standard deviation 7.1 millilitres.

1. Find the probability that 3 randomly chosen cans of cola contain less liquid than 2 randomly chosen bottles of tonic water.
2. A new drink is made by mixing the contents of 2 cans of cola with half a bottle of tonic water. Find the probability that the volume of the new drink is more than 900 millilitres.

2 A computer user finds that unwanted emails arrive randomly at a uniform average rate of 1.27 per hour.

1. Find the probability that more than 1 unwanted email arrives in a period of 5 hours.
2. Find the probability that more than 850 unwanted emails arrive in a period of 700 hours.

3 An airline knows that some people who have bought tickets may not arrive for the flight. The airline therefore sells more tickets than the number of seats that are available. For one flight there are 210 seats available and 213 people have bought tickets. The probability of any person who has bought a ticket not arriving for the flight is \(\frac { 1 } { 50 }\).

1. By considering the number of people who do not arrive for the flight, use a suitable approximation to calculate the probability that more people will arrive than there are seats available. Independently, on another flight for which 135 people have bought tickets, the probability of any person not arriving is \(\frac { 1 } { 75 }\).
2. Calculate the probability that, for both these flights, the total number of people who do not arrive is 5 .

7

1. Random variables \(Y\) and \(X\) are related by \(Y = a + b X\), where \(a\) and \(b\) are constants and \(b > 0\). The standard deviation of \(Y\) is twice the standard deviation of \(X\). The mean of \(Y\) is 7.92 and is 0.8 more than the mean of \(X\). Find the values of \(a\) and \(b\).
2. Random variables \(R\) and \(S\) are such that \(R \sim \mathrm {~N} \left( \mu , 2 ^ { 2 } \right)\) and \(S \sim \mathrm {~N} \left( 2 \mu , 3 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( R + S > 1 ) = 0.9\).
   1. Find \(\mu\).
   2. Hence find \(\mathrm { P } ( S > R )\).

1 A random variable has the distribution \(\mathrm { Po } ( 31 )\). Name an appropriate approximating distribution and state the mean and standard deviation of this approximating distribution.