5.04b Linear combinations: of normal distributions

1. The random variable \(A\) is defined as

$$A = 4 X - 3 Y$$ where \(X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent. Find

1. \(\mathrm { E } ( A )\),
2. \(\operatorname { Var } ( A )\). The random variables \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\) and \(Y _ { 4 }\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
3. Find \(\mathrm { P } ( B > A )\).
   advancing learning, changing lives
   1. A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.
   Test, at the \(5 \%\) level of significance, the managing director's claim. State your hypotheses clearly.
   2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.
4. Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
5. find the probability that Philip beats James in the race by more than 2 minutes.
   3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .
6. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
7. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
8. find a \(98 \%\) confidence interval for \(w\).
   1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.
   advancing learning, changing lives \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-055_2632_1828_123_121}
   2. A county councillor is investigating the level of hardship, h , of a town and the number of calls per 100 people to the emergency services, c. He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
   1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
   \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-081_2642_1833_118_118}
   2. A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than \(5 \%\).
   (5)
   3. The table below shows the population and the number of council employees for different towns and villages. \end{table} A nswers without working may not gain full credit. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{ \(0 - 3\) & 8
   \hline \(3 - 5\) & 12
   \hline \(5 - 6\) & 13
   \hline \(6 - 8\) & 9
   \hline \(8 - 12\) & 8
   \hline \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
9. Show that an estimate of \(\bar { X } = 5.49\) and an estimate of \(S _ { X } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
   Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]

   Waiting Time	\(\mathrm { x } < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(\mathrm { x } > 8\)
   Expected Frequency	8.56	12.73	7.56	a	b

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
10. Find the value of a and the value of b .
11. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.
    \section*{Q uestion 4 continued}
    1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
    One large and 3 small bottles of Blumen are chosen at random.
12. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
13. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
    \section*{Q uestion 5 continued} 6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
14. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
15. Explain the relevance of the Central Limit Theorem to the test in part (a). \section*{Q uestion 6 continued} \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
    7. Lambs are born in a shed on M ill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below. $$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
16. Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
17. Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
18. Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
    \section*{Q uestion 7 continued} \end{figure}

1 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } ( 30,9 )\) and \(\mathrm { N } ( 20,4 )\) respectively.

1. Give the distribution of $$\left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right) - \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } + Y _ { 4 } \right)$$ where \(X _ { i } , i = 1,2,3\), and \(Y _ { j } , j = 1,2,3,4\), are independent observations of \(X\) and \(Y\) respectively. The time for female cadets to complete an assault course is \(X\) minutes and the time for male cadets to complete the same assault course is \(Y\) minutes.
2. Find the probability that the total time for three randomly chosen female cadets to complete the assault course is greater than the total time for four randomly chosen male cadets to complete the assault course.

1 A company hires out narrowboats on a canal. It may be assumed that demands to hire a narrowboat occur independently and randomly at a constant mean rate of 25 per week. Using a suitable normal approximation, find

1. the probability that 15 or fewer narrowboats are hired out during a certain week,
2. the number of narrowboats that the company needs to have available for a week in order that the probability of running out of boats is 0.05 or less.

2

1. The heights of boys in Year 9 are normally distributed with mean 156 cm and standard deviation 8 cm . The heights of girls in Year 10 are, independently, normally distributed with mean 160 cm and standard deviation 7 cm . Find the probability that the mean height of a random sample of 9 boys in Year 9 exceeds the mean height of a random sample of 16 girls in Year 10.
2. State why the distributions of the sample means are normally distributed.

2

1. The statistic \(T\) is derived from a random sample taken from a population which has an unknown parameter \(\theta\). \(T\) is an unbiased estimator of \(\theta\). What does the statement ' \(T\) is an unbiased estimator of \(\theta ^ { \prime }\) imply?
2. A random sample of size \(n\) is taken from each of two independent populations. The first population has a non-zero mean \(\mu\) and variance \(\sigma ^ { 2 }\) and \(\bar { X } _ { 1 }\) denotes the sample mean. The second population has mean \(\frac { 1 } { 2 } \mu\) and variance \(b \sigma ^ { 2 }\), where \(b\) is a positive constant, and \(\bar { X } _ { 2 }\) denotes the sample mean. Two unbiased estimators for \(\mu\) are defined by $$T _ { 1 } = 3 \bar { X } _ { 1 } - a \bar { X } _ { 2 } \quad \text { and } \quad T _ { 2 } = \frac { 1 } { 5 } \left( 4 \bar { X } _ { 1 } + 2 \bar { X } _ { 2 } \right) .$$
   1. Determine the value of \(a\).
   2. Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { \sigma ^ { 2 } } { n } ( 9 + 16 b )\) and find a similar expression for \(\operatorname { Var } \left( T _ { 2 } \right)\).
   3. The estimator with the smaller variance is preferred. State which of \(T _ { 1 }\) and \(T _ { 2 }\) is the preferred estimator of \(\mu\).

4 It is given that \(X\) and \(Y\) are independent random variables with distributions \(\mathrm { N } \left( \mu _ { x } , \sigma _ { x } ^ { 2 } \right)\) and \(\mathrm { N } \left( \mu _ { y } , \sigma _ { y } ^ { 2 } \right)\) respectively, and that \(W\) is a random variable such that \(W = X + Y\).

1. Use moment generating functions to show that the distribution of \(W\) is \(\mathrm { N } \left( \mu _ { x } + \mu _ { y } , \sigma _ { x } ^ { 2 } + \sigma _ { y } ^ { 2 } \right)\).
2. State the distribution of \(X - Y\). The diameters of the central poles of one brand of rotary clothes lines are normally distributed with mean 3.75 cm and variance \(0.000125 \mathrm {~cm} ^ { 2 }\). The diameters of the cylindrical tubes, into which the central poles fit, are normally distributed with mean 3.85 cm and variance \(0.0001 \mathrm {~cm} ^ { 2 }\). Poles and tubes are chosen at random. The 'clearance' between a tube and a pole is the diameter of the tube minus the diameter of the pole.
3. Find the probability that a pole and tube have a clearance between 0.08 cm and 0.13 cm .
4. Given that a pole and tube have a clearance between 0.08 cm and 0.13 cm , find the probability that the clearance is between 0.11 cm and 0.125 cm .

1 The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} ( \mu , 11 ) , \quad Y \sim \mathrm {~N} \left( 10 , \sigma ^ { 2 } \right) \quad \text { and } \quad 2 X - 5 Y \sim \mathrm {~N} ( 0,144 ) .$$ Find

1. the values of \(\mu\) and \(\sigma ^ { 2 }\),
2. \(\mathrm { P } ( X - Y > 10 )\).

2 The mass in grams of a pre-cut piece of Brie cheese is a random variable with the distribution \(\mathrm { N } ( 150,1200 )\). Brie costs 80 p per 100 g .

1. Find the probability that a randomly chosen piece of Brie costs more than \(\pounds 1.40\). The mass in grams of a pre-cut piece of Stilton cheese is an independent random variable with the distribution \(\mathrm { N } ( 180,1500 )\).
2. Find the probability that the total mass of four randomly chosen pieces of Brie is less than the total mass of three randomly chosen pieces of Stilton.

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution \(\operatorname { Po } ( 20 )\).

1. Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.
2. Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .
3. Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.

7 The total mass of a can of pears is the sum of three independent random variables: the mass of pears, the mass of juice, and the mass of the container. The mass in grams of pears in a can has the distribution \(\mathrm { N } ( 300,400 )\). The mass in grams of juice has the distribution \(\mathrm { N } ( 200,60 )\). The mass in grams of the container has the distribution \(\mathrm { N } ( 70,10 )\).

1. Find the probability that the total mass of a randomly chosen can is less than 530 g .
2. Find the probability that the mass of the container of a randomly chosen can is more than one eighth of the total mass of the can.

7 The time taken for me to walk from my house to the bus stop has a normal distribution with mean 10 minutes and standard deviation 1.5 minutes. The arrival time of the bus is normally distributed with mean 0900 and standard deviation 1 minute. If the bus arrives early it does not wait. I leave home at 0845 . Find, correct to 3 decimal places, the probability that I catch the bus.

The time, in minutes, taken by students to complete a test has the distribution \(\text{N}(125, 36)\).

1. Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes. [3]
2. Explain whether it was necessary to use the Central Limit theorem in the solution to part (a). [1]

The masses, in kilograms, of large and small sacks of flour have the distributions \(\text{N}(55, 3^2)\) and \(\text{N}(27, 2.5^2)\) respectively.

1. Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340 kg. Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour. [5]
2. Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour. [5]

The masses, in grams, of small and large bags of flour have the distributions N(510, 100) and N(1015, 324) respectively. André selects 4 small bags of flour and 2 large bags of flour at random.

1. Find the probability that the total mass of these 6 bags of flour is less than 4130 g. [5]
2. Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags. [5]

The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim\) N(10.3, 5.76) and \(Y \sim\) N(11.4, 9.61). The income generated by the chemicals is \\(2.50 per kilogram for \)A\( and \\)3.25 per kilogram for \(B\).

1. Find the mean and variance of the daily income generated by chemical \(A\). [2]
2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\). [6]

The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\text{N}(16.0, 0.4)\) and \(\text{N}(51.0, 0.9)\) 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. [5]

\(X\) is a random variable having the distribution \(\text{B}(12, \frac{1}{4})\). A random sample of 60 values of \(X\) is taken. Find the probability that the sample mean is less than 2.8. [5]

\(X\) and \(Y\) are independent random variables with distributions \(\mathrm{Po}(1.6)\) and \(\mathrm{Po}(2.3)\) respectively.

1. Find \(\mathrm{P}(X + Y = 4)\). [3]

A random sample of 75 values of \(X\) is taken.

1. State the approximate distribution of the sample mean, \(\overline{X}\), including the values of the parameters. [2]
2. Hence find the probability that the sample mean is more than 1.7. [3]
3. Explain whether the Central Limit theorem was needed to answer part (ii). [1]

Bags of sugar are packed in boxes, each box containing 20 bags. The masses of the boxes, when empty, are normally distributed with mean 0.4 kg and standard deviation 0.01 kg. The masses of the bags are normally distributed with mean 1.02 kg and standard deviation 0.03 kg.

1. Find the probability that the total mass of a full box of 20 bags is less than 20.6 kg. [5]
2. Two full boxes are chosen at random. Find the probability that they differ in mass by less than 0.02 kg. [5]

Bottles of wine are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.3 kg and standard deviation 0.06 kg. The weights of the empty racks are normally distributed with mean 2 kg and standard deviation 0.3 kg.

1. Find the probability that the total weight of a full rack of 12 bottles of wine is between 17 kg and 18 kg. [5]
2. Two bottles of wine are chosen at random. Find the probability that they differ in weight by more than 0.05 kg. [5]

Ranjit goes to mathematics lectures and physics lectures. The length, in minutes, of a mathematics lecture is modelled by the variable \(X\) with distribution N(36, 3.5²). The length, in minutes, of a physics lecture is modelled by the independent variable \(Y\) with distribution N(55, 5.2²).

1. Find the probability that the total length of two mathematics lectures and one physics lecture is less than 140 minutes. [4]
2. Ranjit calculates how long he will need to spend revising the content of each lecture as follows. Each minute of a mathematics lecture requires 1 minute of revision and each minute of a physics lecture requires 1½ minutes of revision. Find the probability that the total revision time required for one mathematics lecture and one physics lecture is more than 100 minutes. [4]

The discrete random variable \(X\) is distributed B(\(n\), \(p\)).

1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution. [1]
2. Give a reason to support your value. [1]
3. Given that \(n = 200\) and \(p = 0.48\), find P(\(90 \leq X < 105\)). [7]

A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg [6]

Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.

1. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg [5]

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

1. Write down the distribution of \(M\), the mean weight, in kg, of this sample. [2]
2. Find P(\(M < 78.5\)). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]