5.04b Linear combinations: of normal distributions

5 A manufacturer uses three types of capacitor in a particular electronic device. The capacitances, measured in suitable units, are modelled by independent Normal distributions with means and standard deviations as shown in the table.

1. Determine the probability that the total capacitance of a randomly chosen capacitor of Type B and two randomly chosen capacitors of Type A is at least 16 units.
2. Determine the probability that the capacitance of a randomly chosen capacitor of Type C is within 1 unit of the total capacitance of four randomly chosen capacitors of Type B. When the manufacturer gets a new batch of 1000 capacitors from the supplier, a random sample of 10 of them is tested to check the capacitances. For a new batch of Type C capacitors, summary statistics for the capacitances, \(x\) units, of the random sample are as follows. \(n = 10\) $$\sum x = 299.6 \quad \sum x ^ { 2 } = 8981.0$$ You should assume that the capacitances of the sample come from a Normally distributed population, but you should not assume that the standard deviation is 0.64 as for previous Type C capacitors.
3. In this question you must show detailed reasoning. Carry out a hypothesis test at the \(5 \%\) significance level to check whether it is reasonable to assume that the capacitors in this batch have the specified mean capacitance for Type C of 30.2 units.

5. The masses, \(X\), in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.

1. Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
2. The mean mass, in kg , of a random sample of \(n\) men drawn from this distribution is \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } > 80 )\) is approximately \(0 \cdot 007\), find \(n\). The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
3. A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
4. State a modelling assumption you have made in calculating your answer for part (c).

1. The discrete random variable \(X\) has probability distribution

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 8.79\)
2. find \(\mathrm { E } \left( X ^ { 2 } \right)\) The discrete random variable \(Y\) has probability distribution

   \(y\)	- 2	- 1	0	1	2
   \(\mathrm { P } ( Y = y )\)	\(3 a\)	\(a\)	\(b\)	\(a\)	\(c\)

   where \(a\), \(b\) and \(c\) are constants.
   For the random variable \(Y\) $$\mathrm { P } ( Y \leqslant 0 ) = 0.75 \quad \text { and } \quad \mathrm { E } \left( Y ^ { 2 } + 3 \right) = 5$$
3. Find the value of \(a\), the value of \(b\) and the value of \(c\) The random variable \(W = Y - X\) where \(Y\) and \(X\) are independent.
   The random variable \(T = 3 W - 8\)
4. Calculate \(\mathrm { P } ( W > T )\)

1. A courier delivers parcels. The random variable \(X\) represents the number of parcels delivered successfully each day by the courier where \(X \sim \mathrm {~B} ( 400,0.64 )\)

A random sample \(X _ { 1 } , X _ { 2 } , \ldots X _ { 100 }\) is taken.
Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257

7 A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic. The weights, \(W \mathrm {~kg}\), of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg . The weights, \(X \mathrm {~kg}\), of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg . The random variables \(W\) and \(X\) are independent.

1. Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy. The manufacturer packs \(n\) of these wooden toys and \(2 n\) of these plastic toys into the same container. The maximum weight the container can hold is 252 kg . The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.
2. Calculate the value of \(n\).

7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm . The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm .

1. Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. One large panel and one small panel are selected at random.
2. Find the probability that the length of the large panel is more than \(\frac { 8 } { 3 }\) times the length of the small panel. Rosa needs 1000 cm of fencing. The large panels cost \(\pounds 80\) each and the small panels cost \(\pounds 30\) each. Rosa's plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.
3. Calculate whether or not the expected cost of Rosa's plan is cheaper than simply buying 14 small panels.

1. The weights of a particular type of apple, \(A\) grams, and a particular type of orange, \(R\) grams, each follow independent normal distributions.

$$A \sim \mathrm {~N} \left( 160,12 ^ { 2 } \right) \quad R \sim \mathrm {~N} \left( 140,10 ^ { 2 } \right)$$

1. Find the distribution of
   1. \(A + R\)
   2. the total weight of 2 randomly selected apples. A box contains 4 apples and 1 orange only. Jesse selects 2 pieces of fruit at random from the box.
2. Find the probability that the total weight of the 2 pieces of fruit exceeds 310 grams. From a large number of apples and oranges, Celeste selects \(m\) apples and 1 orange at random. The random variable \(W\) is given by $$W = \left( \sum _ { i = 1 } ^ { m } A _ { i } \right) - n \times R$$ where \(n\) is a positive integer.
   Given that the middle \(95 \%\) of the distribution of \(W\) lies between 1100.08 and 1499.92 grams, (c) find the value of \(m\) and the value of \(n\)

1. The random variable \(X \sim \mathrm {~N} \left( 5,0.4 ^ { 2 } \right)\) and the random variable \(Y \sim \mathrm {~N} \left( 8,0.1 ^ { 2 } \right)\) \(X\) and \(Y\) are independent random variables.
   A random sample of \(a\) independent observations is taken from the distribution of \(X\) and one observation is taken from the distribution of \(Y\)

The random variable \(W = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { a } + b Y\) and has the distribution \(\mathrm { N } \left( 169,2 ^ { 2 } \right)\) Find the value of \(a\) and the value of \(b\)

1. The weights of eggs, \(E\) grams, follow a normal distribution, \(\mathrm { N } \left( 60,3 ^ { 2 } \right)\)

The weights of empty small boxes, \(S\) grams, follow a normal distribution, \(\mathrm { N } \left( 24,1.8 ^ { 2 } \right)\) The weights of empty large boxes, \(L\) grams, follow a normal distribution, \(\mathrm { N } \left( 40,2.1 ^ { 2 } \right)\) Small boxes of eggs contain 6 randomly selected eggs.
Large boxes of eggs contain 12 randomly selected eggs.

1. Find the probability that the total weight of a randomly selected small box of 6 eggs weighs less than 387 grams.
2. Find the probability that a randomly selected large box of 12 eggs weighs more than twice a randomly selected small box of 6 eggs.

1. A company packs chickpeas into small bags and large bags.

The weight of a small bag of chickpeas is normally distributed with mean 500 g and standard deviation 5 g A random sample of 3 small bags of chickpeas is taken.

1. Find the probability that the total weight of these 3 bags of chickpeas is between 1490 g and 1530 g The weight of a large bag of chickpeas is normally distributed with mean 1020 g and standard deviation 20 g One large bag and one small bag of chickpeas are chosen at random.
2. Calculate the probability that the weight of the large bag of chickpeas is at least 30 g more than twice the weight of the small bag of chickpeas. Show your working clearly.

1. Scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.6,0.6 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution N(4.8, 0.32). The random variables \(L\) and \(S\) are independent.

A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Show your working clearly. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.2 )\)

5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by \(X\).

1. \(\mathrm { E } ( X )\),
2. \(\operatorname { Var } ( X )\).
   1. Write down
   2. Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\).
   3. Use the binomial distribution to find exactly the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\). Show the values of all relevant calculations.

9 The continuous random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The sum of a random sample of 16 observations of \(C\) is 224.0 .

1. Find an unbiased estimate of \(\mu\).
2. It is given that an unbiased estimate of \(\sigma ^ { 2 }\) is 0.24. Find the value of \(\Sigma c ^ { 2 }\). \(D\) is the sum of 10 independent observations of \(C\).
3. Explain whether \(D\) has a normal distribution. The continuous random variable \(F\) is normally distributed with mean 15.0, and it is known that \(\mathrm { P } ( F < 13.2 ) = 0.115\).
4. Use the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) to find \(\mathrm { P } ( D + F > 157.0 )\). \section*{OCR} \section*{Oxford Cambridge and RSA}

1 The performance of a piece of music is being recorded. The piece consists of three sections, \(A , B\) and \(C\). The times, in seconds, taken to perform the three sections are normally distributed random variables with the following means and standard deviations.

1. Assume first that the times for the three sections are independent. Find the probability that the total length of the performance is greater than 720.0 seconds.
2. In fact sections \(A\) and \(C\) are musically identical, and the recording is made by using a single performance of section \(A\) twice, together with a performance of section \(B\). In this case find the probability that the total length of the performance is greater than 720.0 seconds.

7

1. The weight of a sack of mixed dog biscuits can be modelled by a normal distribution with a mean of 10.15 kg and a standard deviation of 0.3 kg . A pet shop purchases 12 such sacks that can be considered to be a random sample.
   Calculate the probability that the mean weight of the 12 sacks is less than 10 kg .
2. The weight of dry cat food in a pouch can also be modelled by a normal distribution. The contents, \(x\) grams, of each of a random sample of 40 pouches were weighed. Subsequent analysis of these weights gave $$\bar { x } = 304.6 \quad \text { and } \quad s = 5.37$$
   1. Construct a \(99 \%\) confidence interval for the mean weight of dry cat food in a pouch. Give the limits to one decimal place.
   2. Comment, with justification, on each of the following two claims. Claim 1: The mean weight of dry cat food in a pouch is more than 300 grams.
      Claim 2: All pouches contain more than 300 grams of dry cat food.
      [0pt] [4 marks]
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-24_2288_1705_221_155}

4 The table below shows the probability distribution for the number of students, \(R\), attending classes for a particular mathematics module.

1. Find values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
2. The number of students, \(S\), attending classes for a different mathematics module is such that $$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$ Find values for the mean and variance of:
   1. \(T = R + S\);
   2. \(\quad D = S - R\).

5 The duration, \(X\) minutes, of a timetabled 1-hour lesson may be assumed to be normally distributed with mean 54 and standard deviation 2. The duration, \(Y\) minutes, of a timetabled \(1 \frac { 1 } { 2 }\)-hour lesson may be assumed to be normally distributed with mean 83 and standard deviation 3. Assuming the durations of lessons to be independent, determine the probability that the total duration of a random sample of three 1 -hour lessons is less than the total duration of a random sample of two \(1 \frac { 1 } { 2 }\)-hour lessons.
(7 marks)

2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.

1. One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
3. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
4. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac { 1 } { 4 } X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.
5. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
6. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac { 3 } { 5 } T\). Find the probability as given by this model that a question can be prepared in less than \(1 \frac { 3 } { 4 }\) hours.

5 A particular brand of pasta is sold in bags of two different sizes. The mass of pasta in the large bags is advertised as being 1500 g ; in fact it is Normally distributed with mean 1515 g and standard deviation 4.7 g . The mass of pasta in the small bags is advertised as being 500 g ; in fact it is Normally distributed with mean 508 g and standard deviation 3.3 g .

1. Find the probability that the total mass of pasta in 5 randomly selected small bags is less than 2550 g .
2. Find the probability that the mass of pasta in a randomly selected large bag is greater than three times the mass of pasta in a randomly selected small bag.

11 Two girls, Lili and Hui, play a game with a fair six-sided dice. The dice is thrown 10 times. \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) represent the scores on the \(1 ^ { \text {st } } , 2 ^ { \text {nd } } , \ldots , 10 ^ { \text {th } }\) throws of the dice. \(L\) denotes Lili's score and \(L = 10 X _ { 1 }\). \(H\) denotes Hui's score and \(H = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).

1. Calculate
   The spreadsheet below shows a simulation of 25 plays of the game. The cell E3, highlighted, shows the score when the dice is thrown the fourth time in the first game. \begin{table}[h]

   	A	B	C	D	E	F	G	H	I	J	K	L	M	N
   1	Throw of dice												Lili's	Hui's
   2		1	2	3	4	5	6	7	8	9	10		score	score
   3	Game 1	3	5	2	1	1	3	1	1	1	4		30	22
   4	Game 2	6	3	2	4	4	3	5	3	3	5		60	38
   5	Game 3	6	4	2	6	5	2	1	5	2	3		60	36
   6	Game 4	1	5	1	6	6	3	1	4	6	2		10	35
   7	Game 5	4	4	3	1	6	4	4	1	6	2		40	35
   8	Game 6	2	1	5	1	2	5	1	5	2	3		20	27
   9	Game 7	1	1	3	4	4	5	6	3	4	2		10	33
   10	Game 8	1	1	3	6	3	4	4	5	2	3		10	32
   11	Game 9	2	2	2	4	3	2	1	5	5	6		20	32
   12	Game 10	3	5	3	3	5	3	4	3	1	1		30	31
   13	Game 11	5	3	6	5	5	4	2	1	1	5		50	37
   14	Game 12	6	4	3	2	4	1	3	3	5	3		60	34
   15	Game 13	2	3	2	1	2	2	2	2	2	1		20	19
   16	Game 14	4	1	3	3	1	2	6	6	1	3		40	30
   17	Game 15	5	1	2	6	3	4	6	3	6	4		50	40
   18	Game 16	3	6	1	1	5	3	1	3	3	3		30	29
   19	Game 17	5	2	5	2	4	5	2	2	3	4		50	34
   20	Game 18	3	6	3	5	5	2	3	1	1	2		30	31
   21	Game 19	6	6	3	1	5	6	3	4	1	6		60	41
   22	Game 20	2	6	4	5	6	5	2	4	3	3		20	40
   23	Game 21	5	3	5	4	5	3	3	6	6	1		50	41
   24	Game 22	6	3	5	5	6	3	5	6	1	1		60	41
   25	Game 23	5	4	5	5	6	4	2	1	3	6		50	41
   26	Game 24	3	5	2	3	2	4	3	2	3	3		30	30
   27	Game 25	5	2	4	2	4	5	2	2	5	2		50	33
   28
   29												mean	37.60	33.68
   30												sd	17.39	5.77

   \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{table}
2. Use the simulation to estimate \(\mathrm { P } ( L > 40 )\) and \(\mathrm { P } ( H > 40 )\).
3. (A) Calculate the exact value of \(\mathrm { P } ( L > 40 )\).
   (B) Comment on how the exact value compares with your estimate of \(\mathrm { P } ( L > 40 )\) in part (v). Hui wonders whether it is appropriate to use the Central Limit Theorem to approximate the distribution of \(X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).
4. (A) State what type of diagram Hui could draw, based on the output from the spreadsheet, to investigate this.
   (B) Explain how she should interpret the diagram.
5. (A) Calculate an approximate value of \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 } > 40 \right)\) using the Central Limit Theorem.
   (B) Comment on how this value compares with your estimate of \(\mathrm { P } ( H > 40 )\) in part (v). \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
   For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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2. Geraint is a beekeeper. The amounts of honey, \(X \mathrm {~kg}\), that he collects annually, from each hive are modelled by the normal distribution \(\mathrm { N } \left( 15,5 ^ { 2 } \right)\). At location \(A\), Geraint has three hives and at location \(B\) he has five hives. You may assume that the amounts of honey collected from the eight hives are independent of each other.

1. 1. Find the probability that Geraint collects more than 14 kg of honey from the first hive at location \(A\).
   2. Find the probability that he collects more than 14 kg of honey from exactly two out of the three hives at location \(A\).
2. Find the probability that the total amount of honey that Geraint collects from all eight hives is more than 160 kg .
3. Find the probability that Geraint collects at least twice as much honey from location B as from location A.

7. \includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral \(A B C D\), where \(\widehat { B A D } = \alpha , \widehat { B C D } = \beta\) and \(\alpha + \beta = 180 ^ { \circ }\). These angles are measured.
The random variables \(X\) and \(Y\) denote the measured values, in degrees, of \(\widehat { B A D }\) and \(\widehat { B C D }\) respectively. You are given that \(X\) and \(Y\) are independently normally distributed with standard deviation \(\sigma\) and means \(\alpha\) and \(\beta\) respectively.

1. Calculate, correct to two decimal places, the probability that \(X + Y\) will differ from \(180 ^ { \circ }\) by less than \(\sigma\).
2. Show that \(T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )\) is an unbiased estimator for \(\alpha\) and verify that it is a better estimator than \(X\) for \(\alpha\).
3. Now consider \(T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)\).
   1. Show that \(T _ { 2 }\) is an unbiased estimator for \(\alpha\) for all values of \(\lambda\).
   2. Find \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(\sigma\).
   3. Hence determine the value of \(\lambda\) which gives the best unbiased estimator for \(\alpha\).

1 The performance of a piece of music is being recorded. The piece consists of three sections, \(A , B\) and \(C\). The times, in seconds, taken to perform the three sections are normally distributed random variables with the following means and standard deviations. \end{table}

7. A set of scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution \(\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)\). The random variables \(L\) and \(S\) are independent. A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.1 )\).
   \end{table}
   1. Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, \(x \mathrm {~mm}\) of each wading bird was recorded. The results are summarised as follows
   $$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$
4. Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
5. find a \(99 \%\) confidence interval for the mean wing length of the birds from this group.
   2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results \end{table}
   1. A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
   2. Explain in detail how the researcher should obtain such a sample.
   3. Give one advantage and one disadvantage of
      1. quota sampling,
      2. systematic sampling.
   4. The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm . The heights of the orchids are normally distributed.
   Given that the population standard deviation is 0.5 cm ,
6. estimate limits between which \(95 \%\) of the heights of the orchids lie,
7. find a 98\% confidence interval for the mean height of the orchids. A grower claims that the mean height of this type of orchid is 19.5 cm .
8. Comment on the grower's claim. Give a reason for your answer.
   3. A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.

   Individual	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
   BMI	17.4	21.4	18.9	24.4	19.4	20.1	22.6	18.4	25.8	28.1
   Finishing position	3	5	1	9	6	4	10	2	7	8

9. Calculate Spearman's rank correlation coefficient for these data.
10. Stating your hypotheses clearly and using a one tailed test with a \(5 \%\) level of significance, interpret your rank correlation coefficient.
11. Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.
    4. A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.
    5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below. \begin{table}[h]

    Number of goals	Frequency
    0	40
    1	33
    2	14
    3	8
    4	5

    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
12. Calculate the mean number of goals scored per game. The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]

    Number of goals	Expected Frequency
    0	34.994
    1	\(r\)
    2	\(s\)
    3	6.752
    \(\geqslant 4\)	2.221

    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
13. Find the value of \(r\) and the value of \(s\) giving your answers to 3 decimal places.
14. Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim.
    1. The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm . The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm .
    2. Test, using a \(5 \%\) level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.
    3. State two assumptions you made in carrying out the test in part (a).
       
    4. A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below.
       119.9
       120.3
       120.1
       120.4
       120.2
    5. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.
    The lengths of climbing rope are known to have a standard deviation of 0.2 m . The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of \(n\), lies within 0.05 m of its true value.
15. Find the minimum sample size required.