Comparison involving sums or multiples

2 Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions \(W \sim \mathrm {~N} ( 1520,450 ) , X \sim \mathrm {~N} ( 2250,720 )\) and \(Y \sim \mathrm {~N} ( 3860,1050 )\). Find the probability that, on a particular day, \(Y\) is less than the total of \(W\) and \(X\).

3 The lengths, in centimetres, of two types of insect, \(A\) and \(B\), are modelled by the random variables \(X \sim \mathrm {~N} ( 6.2,0.36 )\) and \(Y \sim \mathrm {~N} ( 2.4,0.25 )\) respectively. Find the probability that the length of a randomly chosen type \(A\) insect is greater than the sum of the lengths of 3 randomly chosen type \(B\) insects.

3 The lengths of red pencils are normally distributed with mean 6.5 cm and standard deviation 0.23 cm .

1. Two red pencils are chosen at random. Find the probability that their total length is greater than 12.5 cm . The lengths of black pencils are normally distributed with mean 11.3 cm and standard deviation 0.46 cm .
2. Find the probability that the total length of 3 red pencils is more than 6.7 cm greater than the length of 1 black pencil.

6 The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \(\mathrm { N } \left( 78.8,12.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 62.0,10.0 ^ { 2 } \right)\) respectively.

1. The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load.
2. Find the probability that a randomly chosen carton of sugar has a smaller mass than a randomly chosen carton of flour.

6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.

1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man. [4]

4 The weights of men follow a normal distribution with mean 71 kg and standard deviation 7 kg . The weights of women follow a normal distribution with mean 57 kg and standard deviation 5 kg . The total weight of 5 men and 2 women chosen randomly is denoted by \(X \mathrm {~kg}\).

1. Show that \(\mathrm { E } ( X ) = 469\) and \(\operatorname { Var } ( X ) = 295\).
2. The total weight of 4 men and 3 women chosen randomly is denoted by \(Y \mathrm {~kg}\). Find the mean and standard deviation of \(X - Y\) and hence find \(\mathrm { P } ( X - Y > 22 )\).

5 Climbing ropes produced by a manufacturer have breaking strengths which are normally distributed with mean 160 kg and standard deviation 11.3 kg . A group of climbers have weights which are normally distributed with mean 66.3 kg and standard deviation 7.1 kg .

1. Find the probability that a rope chosen randomly will break under the combined weight of 2 climbers chosen randomly. Each climber carries, in a rucksack, equipment amounting to half his own weight.
2. Find the mean and variance of the combined weight of a climber and his rucksack.
3. Find the probability that the combined weight of a climber and his rucksack is greater than 87 kg .

6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.

1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man.

4 Cola is sold in bottles and cans. The volume of cola in a bottle is normally distributed with mean 500 ml and standard deviation 10 ml . The volume of cola in a can is normally distributed with mean 330 ml and standard deviation 8 ml . Find the probability that the total volume of cola in 2 randomly selected bottles is greater than 3 times the volume of cola in a randomly selected can.

1. Small containers and large containers are independently filled with fruit juice.

The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable \(W\) represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container. \(W \sim \mathrm {~N} ( a , b )\) where \(a\) and \(b\) are positive constants.

1. Find the value of \(a\) and the value of \(b\)
2. Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container. A random sample of 3 small containers of fruit juice is taken.
3. Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.

7. The heights, in cm, of the male employees in a large company follow a normal distribution with mean 177 and standard deviation 5 The heights, in cm, of the female employees follow a normal distribution with mean 163 and standard deviation 4 A male employee and a female employee are chosen at random.

1. Find the probability that the male employee is taller than the female employee. Six male employees and four female employees are chosen at random.
2. Find the probability that their total height is less than 17 m .

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.

1. 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.
2. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.

7. Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.

1. Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.
2. Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar. A random sample of 5 medium bags of sugar is taken.
3. Find the value of \(d\) so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than \(d\) grams.

6 The length \(L\) of a particular type of fence panel is Normally distributed with mean 179.2 cm and standard deviation 0.8 cm . You should assume that the lengths of individual fence panels are independent of each other.

1. Find the probability that the length of a randomly chosen fence panel is at least 180 cm .
2. Find the probability that the total length of 5 randomly chosen fence panels is less than 895 cm . The width \(W\) of a fence post is Normally distributed with mean 9.8 cm and standard deviation 0.3 cm . A straight fence is constructed using 6 posts and 5 panels with no gaps between them. Fig. 6 shows a view from above of the first two posts, the first panel and the start of the second panel. You should assume that the lengths of fence panels and widths of fence posts are independent. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4caa7409-cb32-41da-ad64-012a45753296-6_213_1522_934_244} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
3. Determine the probability that the total length of the fence, including the posts, is less than 9.5 m .
4. State another assumption that is necessary for the calculation of the probability in part (c) to be valid.

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.

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.

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.

The workers in a large office block use a lift that can carry a maximum load of 1090 kg. The weights of the male workers are normally distributed with mean 78.5 kg and standard deviation 12.6 kg. The weights of the female workers are normally distributed with mean 62.0 kg and standard deviation 9.8 kg. Random samples of 7 males and 8 females can enter the lift.

1. Find the mean and variance of the total weight of the 15 people that enter the lift. [4]
2. Comment on any relationship you have assumed in part (a) between the two samples. [1]
3. Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people. [4]

The random variable \(A\) is defined as $$A = 4X - 3Y$$ where \(X \sim \text{N}(30, 3^2)\), \(Y \sim \text{N}(20, 2^2)\) and \(X\) and \(Y\) are independent. Find

1. E(\(A\)), [2]
2. Var(\(A\)). [3]

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 $$B = \sum_{i=1}^{4} Y_i$$

1. Find P(\(B > A\)). [6]

An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.

1. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms. [3]
2. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms. [3]
3. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms. [3]
4. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor. [5]
5. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a 95\% confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified. [4]