5.04b Linear combinations: of normal distributions

Aram is a packer at a supermarket checkout and the time he takes to pack a randomly chosen item has mean 1.5 s and standard deviation 0.4 s . Justifying any approximation that you make, find the probability that Aram will pack 50 randomly chosen items in less than 70 s . Find the greatest number of items that Aram could pack within 70 s with probability at least \(90 \%\). Huldu is also a packer at the supermarket. The time that she takes to pack a randomly chosen item has mean 1.3 s and standard deviation 0.5 s . Aram and Huldu each have 50 items to pack. Find the probability that Huldu takes a shorter time than Aram.

2 In the vegetable section of a local supermarket, leeks are on sale either loose (and unprepared) or prepared in packs of 4 . The weights of unprepared leeks are modelled by the random variable \(X\) which has the Normal distribution with mean 260 grams and standard deviation 24 grams. The prepared leeks have had \(40 \%\) of their weight removed, so that their weights, \(Y\), are modelled by \(Y = 0.6 X\).

1. Find the probability that a randomly chosen unprepared leek weighs less than 300 grams.
2. Find the probability that a randomly chosen prepared leek weighs more than 175 grams.
3. Find the probability that the total weight of 4 randomly chosen prepared leeks in a pack is less than 600 grams.
4. What total weight of prepared leeks in a randomly chosen pack of 4 is exceeded with probability 0.975 ?
5. Sandie is making soup. She uses 3 unprepared leeks and 2 onions. The weights of onions are modelled by the Normal distribution with mean 150 grams and standard deviation 18 grams. Find the probability that the total weight of her ingredients is more than 1000 grams.
6. A large consignment of unprepared leeks is delivered to the supermarket. A random sample of 100 of them is taken. Their weights have sample mean 252.4 grams and sample standard deviation 24.6 grams. Find a \(99 \%\) confidence interval for the true mean weight of the leeks in this consignment.

9

1. The masses, in grams, of plums of a certain kind have the distribution \(\mathrm { N } ( 55,18 )\).
   1. Find the probability that a plum chosen at random has a mass between 50.0 and 60.0 grams.
   2. The heaviest \(5 \%\) of plums are classified as extra large. Find the minimum mass of extra large plums.
   3. The plums are packed in bags, each containing 10 randomly selected plums. Find the probability that a bag chosen at random has a total mass of less than 530 g .
2. The masses, in grams, of apples of a certain kind have the distribution \(\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)\). It is given that half of the apples have masses between 62 g and 72 g . Determine \(\sigma\).

10 The screenshot in Fig. 10 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h] \end{figure} The area of each of the unshaded regions under the curve is 0.025 . The lower boundary of the shaded region is at 16.452 and the upper boundary of the shaded region is at 21.548 .

1. Calculate the value of \(\mu\).
2. Calculate the value of \(\sigma ^ { 2 }\).
3. \(Y\) is the random variable given by \(Y = 4 X + 5\).
   (A) Write down the distribution of \(Y\).
   (B) Find \(\mathrm { P } ( \mathrm { Y } > 90 )\).

5 A company uses two drivers for deliveries.
Driver \(A\) charges a fixed rate of \(\pounds 80\) per day plus \(\pounds 2\) per mile travelled on that day. Driver \(B\) charges a fixed rate of \(\pounds 120\) per day plus \(\pounds 1.50\) per mile travelled on that day.
On each working day the total distance, in miles, travelled by each driver is a random variable with the distribution \(\mathrm { N } ( 83,360 )\).

1. Find the probability that driver \(A\) charges the company less than \(\pounds 235.00\) for a randomly chosen day's deliveries.
2. Find the probability that the total charge to the company of three randomly chosen days' deliveries by driver \(A\) is at least \(\pounds 300\) more than the total charge of two randomly chosen days' deliveries by driver \(B\).

2 The mass \(J \mathrm {~kg}\) of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass Kkg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04 .

1. Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg .
2. Find the probability that the mass of one bag of King Edward potatoes is more than \(75 \%\) of the mass of one bag of Jersey potatoes.

5. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2 \\ k ( x - 2 ) , & x = 3 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = 0.25\).
2. Find \(\mathrm { E } ( X )\) and show that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.5\).
3. Find \(\operatorname { Var } ( 3 X - 2 )\). Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
4. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\).
5. Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
6. Find \(\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)\).

3. When Rohit plays a game, the number of points he receives is given by the discrete random variable \(X\) with the following probability distribution.

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { F } ( 1.5 )\).
3. Show that \(\operatorname { Var } ( X ) = 1\)
4. Find \(\operatorname { Var } ( 5 - 3 X )\). Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.
5. Find the probability that Rohit wins the prize.

1. The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.

Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)

6

1. Explain what you understand by the sampling distribution of a statistic. At Sam's cafe a standard breakfast consists of 6 breakfast items. Customers can then choose to upgrade to a medium breakfast by adding 1 extra breakfast item or they can upgrade to a large breakfast by adding 2 extra breakfast items. Standard, medium and large breakfasts are sold in the ratio \(6 : 3 : 2\) respectively. A random sample of 2 customers is taken from customers who have bought a breakfast from Sam's cafe on a particular day.
2. Find the sampling distribution for the total number, \(T\), of breakfast items bought by these 2 customers. Show your working clearly.
3. Find \(\mathrm { E } ( T )\)

1. A multiple choice examination paper has \(n\) questions where \(n > 30\)

Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.
7.

3. An airline knows that overall \(3 \%\) of passengers do not turn up for flights. The airline decides to adopt a policy of selling more tickets than there are seats on a flight. For an aircraft with 196 seats, the airline sold 200 tickets for a particular flight.

1. Write down a suitable model for the number of passengers who do not turn up for this flight after buying a ticket. By using a suitable approximation, find the probability that
2. more than 196 passengers turn up for this flight,
3. there is at least one empty seat on this flight.

7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key \(T\) s after the star first appears, a simple model of the game assumes that \(T\) is a continuous uniform random variable defined over the interval \([ 0,1 ]\).

1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
2. Write down E(T).
3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5 \\ 4 - 4 t , & 0.5 \leq t \leq 1 , \\ 0 , & \text { otherwise } . \end{cases}$$
6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
   (4) END

3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.

1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
2. a randomly chosen page has no misprints,
3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.

5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.

1. Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
2. Estimate the probability there are less than 5 defective articles.

An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.

1. In a document of 2000 words find the probability that the administrator makes 4 or more errors. The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.
2. Use a suitable approximation to calculate the probability that the report is accepted.

5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .

1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.

A shopkeeper knows, from past records, that \(15 \%\) of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.

1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till. During the refurbishment a new sandwich display was installed. Before the refurbishment \(20 \%\) of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.
2. Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper's claim. Use a \(10 \%\) level of significance.
   

7. As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable \(X\).

1. Suggest a suitable distribution for \(X\).
   (2) Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable \(S\) represents the final score, in points, for an applicant who chooses answers to this test at random.
2. Show that \(S = 5 X - 20\)
3. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\). An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.
4. Find \(\mathrm { P } ( S \geqslant 20 )\) (4) Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4
5. Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.

1. A cadet fires shots at a target at distances ranging from 25 m to 90 m . The probability of hitting the target with a single shot is \(p\). When firing from a distance \(d \mathrm {~m} , p = \frac { 3 } { 200 } ( 90 - d )\). Each shot is fired independently.

The cadet fires 10 shots from a distance of 40 m .

1. 1. Find the probability that exactly 6 shots hit the target.
   2. Find the probability that at least 8 shots hit the target. The cadet fires 20 shots from a distance of \(x \mathrm {~m}\).
2. Find, to the nearest integer, the value of \(x\) if the cadet has an \(80 \%\) chance of hitting the target at least once. The cadet fires 100 shots from 25 m .
3. Using a suitable approximation, estimate the probability that at least 95 of these shots hit the target.
   

5.

1. State the conditions under which the normal distribution may be used as an approximation to the binomial distribution. A company sells seeds and claims that 55\% of its pea seeds germinate.
2. Write down a reason why the company should not justify their claim by testing all the pea seeds they produce. To test the company's claim, a random sample of 220 pea seeds was planted.
3. State the hypotheses for a two-tailed test of the company's claim. Given that 135 of the 220 pea seeds germinated,
4. use a normal approximation to test, at the \(5 \%\) level of significance, whether or not the company's claim is justified.
   

5. In a large school, \(20 \%\) of students own a touch screen laptop. A random sample of \(n\) students is chosen from the school. Using a normal approximation, the probability that more than 55 of these \(n\) students own a touch screen laptop is 0.0401 correct to 3 significant figures. Find the value of \(n\).
(8)
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6. A potter makes decorative tiles in two colours, red and yellow. The length, \(R \mathrm {~cm}\), of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm . The length, \(Y \mathrm {~cm}\), of the yellow tiles has the normal distribution \(\mathrm { N } \left( 12,0.8 ^ { 2 } \right)\). The random variables \(R\) and \(Y\) are independent. A red tile and a yellow tile are chosen at random.

1. Find the probability that the yellow tile is longer than the red tile. Taruni buys 3 red tiles and 1 yellow tile.
2. Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile. Stefan defines the random variable \(X = a R + b Y\), where \(a\) and \(b\) are constants. He wants to use values of \(a\) and \(b\) such that \(X\) has a mean of 780 and minimum variance.
3. Find the value of \(a\) and the value of \(b\) that Stefan should use. \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-19_2255_50_314_34}

1. A market stall sells vegetables. Two of the vegetables sold are broccoli heads and cabbages.

The weights of these broccoli heads, \(B\) kilograms, follow a normal distribution $$B \sim \mathrm {~N} \left( 0.588,0.084 ^ { 2 } \right)$$ The weights of these cabbages, \(C\) kilograms, follow a normal distribution $$C \sim \mathrm {~N} \left( 0.908,0.039 ^ { 2 } \right)$$

1. Find the probability that the total weight of two randomly chosen broccoli heads is less than the weight of a randomly chosen cabbage. Broccoli heads cost \(\pounds 2.50\) per kg and cabbages cost \(\pounds 3.00\) per kg. Jaymini buys 1 broccoli head and 2 cabbages, chosen randomly.
2. Find the probability that she pays more than £7 The market stall offers a discount for buying 5 or more broccoli heads. The price with the discount is \(\pounds w\) per kg. Let \(\pounds D\) be the price with the discount of 5 broccoli heads.
3. Find, in terms of \(w\), the mean and standard deviation of \(D\) Given that \(\mathrm { P } ( D < 6 ) < 0.1\)
4. find the smallest possible value of \(w\), giving your answer to 2 decimal places.

1. Charlie is training for three events: a 1500 m swim, a 40 km bike ride and a 10 km run.

From past experience his times, in minutes, for each of the three events independently have the following distributions. $$\begin{aligned} & S \sim \mathrm {~N} \left( 41,5.2 ^ { 2 } \right) \text { represents the time for the swim } \\ & B \sim \mathrm {~N} \left( 81,4.2 ^ { 2 } \right) \text { represents the time for the bike ride } \\ & R \sim \mathrm {~N} \left( 57,6.6 ^ { 2 } \right) \text { represents the time for the run } \end{aligned}$$

1. Find the probability that Charlie's total time for a randomly selected swim, bike ride and run exceeds 3 hours.
2. Find the probability that the time for a randomly selected swim will be at least 20 minutes quicker than the time for a randomly selected run. Given that \(\mathrm { P } ( S + B + R > t ) = 0.95\)
3. find the value of \(t\) A triathlon consists of a 1500 m swim, immediately followed by a 40 km bike ride, immediately followed by a 10 km run. Charlie uses the answer to part (a) to find the probability that, in 6 successive independent triathlons, his time will exceed 3 hours on at least one occasion.
4. Find the answer Charlie should obtain. Jane says that Charlie should not have used the answer to part (a) for the calculation in part (d).
5. Explain whether or not Jane is correct.