1. A machine produces packets of sugar. The weights in grams of thirty packets chosen at random are shown below.

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1 No Gdc 1. A machine produces packets of sugar. The weights in grams of thirty packets chosen at random are shown below. Weight (g) Frequency Find unbiased estimates of the mean of the population from which this sample is taken; the variance of the population from which this sample is taken.. Consider the six numbers,, 3, 6, 9, a and b. The mean of the numbers is 6 and the variance is 10. Find the value of a and of b, if a < b. 3. A continuous random variable, X, has probability density function (Total 3 marks) sin x, 0 x π. Find the median of X. 4. The table below shows the probability distribution of a discrete random variable X. x P(X = x) 0. a b 0.5 Given that E(X) = 1.55, find the value of a and of b. Calculate Var(X). 5. A random sample drawn from a large population contains the following data Calculate an unbiased estimate of 6., 7.8, 1.1, 9.7, 5., 14.8, 16., 3.7. the population mean; the population variance. 1

2 6. In a sample of 50 boxes of light bulbs, the number of defective light bulbs per box is shown below. Number of defective light bulbs per box Number of boxes Calculate the median number of defective light bulbs per box. Calculate the mean number of defective light bulbs per box. 7. The local Football Association consists of ten teams. Team A has a 40% chance of winning any game against a higher-ranked team, and a 75% chance of winning any game against a lower-ranked team. If A is currently in fourth position, find the probability that A wins its next game. 8. The box-and-whisker plots shown represent the heights of female students and the heights of male students at a certain school. (Total 4 marks) Females Males Height (cm) What percentage of female students are shorter than any male students? What percentage of male students are shorter than some female students? From the diagram, estimate the mean height of the male students. 9. Bag 1 contains 4 red cubes and 5 blue cubes. Bag contains 7 red cubes and blue cubes. Two cubes are drawn at random, the first from Bag 1 and the second from Bag. (Total 3 marks) Find the probability that the cubes are of the same colour. Given that the cubes selected are of different colours, find the probability that the red cube was selected from Bag Box A contains 6 red balls and green balls. Box B contains 4 red balls and 3 green balls. A fair cubical die with faces numbered 1,, 3, 4, 5, 6 is thrown. If an even number is obtained, a ball is selected from box A; if an odd number is obtained, a ball is selected from box B. Calculate the probability that the ball selected was red.

3 Given that the ball selected was red, calculate the probability that it came from box B. 11. A factory has a machine designed to produce 1 kg bags of sugar. It is found that the average weight of sugar in the bags is 1.0 kg. Assuming that the weights of the bags are normally distributed, find the standard deviation if 1.7% of the bags weigh below 1 kg. Give your answer correct to the nearest 0.1 gram. 1. The random variable X is distributed normally with mean 30 and standard deviation. Find p(7 X 34). 13. The continuous random variable X has probability density function f (x) where (Total 4 marks) (Total 4 marks) f k (x) = e 0, ke kx, 0 x 1 otherwise Show that k = 1. What is the probability that the random variable X has a value that lies between 1 and 1? Give your answer in terms of e. 4 Find the mean and variance of the distribution. Give your answers exactly, in terms of e. (6) The random variable X above represents the lifetime, in years, of a certain type of battery. (d) Find the probability that a battery lasts more than six months. A calculator is fitted with three of these batteries. Each battery fails independently of the other two. Find the probability that at the end of six months (e) (f) none of the batteries has failed; exactly one of the batteries has failed. (Total 17 marks) 14. A machine is set to produce bags of salt, whose weights are distributed normally, with a mean of 110 g and standard deviation of 1.14 g. If the weight of a bag of salt is less than 108 g, the bag is rejected. With these settings, 4% of the bags are rejected. The settings of the machine are altered and it is found that 7% of the bags are rejected. (i) If the mean has not changed, find the new standard deviation, correct to three decimal places. The machine is adjusted to operate with this new value of the standard deviation. 3

4 (ii) Find the value, correct to two decimal places, at which the mean should be set so that only 4% of the bags are rejected. With the new settings from part, it is found that 80% of the bags of salt have a weight which lies between A g and B g, where A and B are symmetric about the mean. Find the values of A and B, giving your answers correct to two decimal places. (Total 1 marks) 15. The lifetime of a particular component of a solar cell is Y years, where Y is a continuous random variable with probability density function 0 y) - 0.5e f ( y / when y 0 when y 0. Find the probability, correct to four significant figures, that a given component fails within six months. Each solar cell has three components which work independently and the cell will continue to run if at least two of the components continue to work. Find the probability that a solar cell fails within six months. (Total 7 marks) 16. The diameters of discs produced by a machine are normally distributed with a mean of 10 cm and standard deviation of 0.1 cm. Find the probability of the machine producing a disc with a diameter smaller than 9.8 cm. 17. Z is the standardized normal random variable with mean 0 and variance 1. Find the value of a such that P( Z a) = A continuous random variable X has probability density function (Total 3 marks) (Total 3 marks) 4 f ( x) (1 x 0,, ) for0 x 1, elsewhere Find E(X). 19. The weights of a certain species of bird are normally distributed with mean 0.8 kg and standard deviation 0.1 kg. Find the probability that the weight of a randomly chosen bird of the species lies between 0.74 kg and 0.95 kg. 0. The probability density function f (x), of a continuous random variable X is defined by (Total 3 marks) 4

5 1 x(4 x ), 0 x 4 0, otherwise. Calculate the median value of X. 1. At a building site the probability, P(A), that all materials arrive on time is The probability, P(B), that the building will be completed on time is The probability that the materials arrive on time and that the building is completed on time is (i) (ii) Show that events A and B are not independent. All the materials arrive on time. Find the probability that the building will not be completed on time. There was a team of ten people working on the building, including three electricians and two plumbers. The architect called a meeting with five of the team, and randomly selected people to attend. Calculate the probability that exactly two electricians and one plumber were called to the meeting. The number of hours a week the people in the team work is normally distributed with a mean of 4 hours. 10% of the team work 48 hours or more a week. Find the probability that both plumbers work more than 40 hours in a given week. (8) (Total 15 marks). The random variable X is normally distributed and Find E(X). P(X 10) = P(X 1) = A random variable X is normally distributed with mean and standard deviation σ, such that P(X > 50.3) = 0.119, and P(X < 43.56) = Find and. Hence find P( X < 5). (Total 7 marks) 4. The following diagram shows the probability density function for the random variable X, which is normally distributed with mean 50 and standard deviation 50. 5

6 fx () x Find the probability represented by the shaded region. 5. The discrete random variable X has the following probability distribution. P(X = x) = k, x 1,, 3, 4 x 0, otherwise Calculate the value of the constant k; E(X). 6. Let f (x) be the probability density function for a random variable X, where f (x) kx 0,,for0 x otherwise Show that k = 8 3. Calculate (i) E(X); (ii) the median of X. (6) (Total 8 marks) 7. A continuous random variable X has probability density function given by Find the value of k. Find P(0.5 x 0.5). k (x x ), for 0 x 0, elsewhere. 6

7 8. Ian and Karl have been chosen to represent their countries in the Olympic discus throw. Assume that the distance thrown by each athlete is normally distributed. The mean distance thrown by Ian in the past year was m with a standard deviation of 1.95 m. In the past year, 80% of Ian s throws have been longer than x metres. Find x, correct to two decimal places. In the past year, 80% of Karl s throws have been longer than 56.5 m. If the mean distance of his throws was m, find the standard deviation of his throws, correct to two decimal places. This year, Karl s throws have a mean of m and a standard deviation of 3.00 m. Ian s throws still have a mean of m and standard deviation 1.95 m. In a competition an athlete must have at least one throw of 65 m or more in the first round to qualify for the final round. Each athlete is allowed three throws in the first round. (i) Determine which of these two athletes is more likely to qualify for the final on their first throw. (ii) Find the probability that both athletes qualify for the final. (11) (Total 17 marks) 9. The speeds of cars at a certain point on a straight road are normally distributed with mean μ and standard deviation σ. 15% of the cars travelled at speeds greater than 90 km h 1 and 1% of them at speeds less than 40 km h 1. Find μ and σ. 30. The continuous random variable X has probability density function 6 1 x(1 + x ) for 0 x, 0 otherwise. Sketch the graph of f for 0 x. Write down the mode of X. Find the mean of X. (d) Find the median of X. 31. Use mathematical induction to prove that 5 n + 9 n + is divisible by 4, for n +. (1) (Total 1 marks) (Total 9 marks) 3. The probability density function f (x) of the continuous random variable X is defined on the interval [0, a] by 7

8 1 x f ( x) 8 7 8x for 0 x 3. for 3 x a. Find the value of a. 33. A continuous random variable X has the probability density function f given by 8, 0 x π ( x 4) 0, otherwise. State the mode of X. Find the exact value of E (X). 34. A continuous random variable X has probability density function f defined by x e, 0, for0 xln otherwise. Find the exact value of E(X). 35. The random variable T has the probability density function t f (t) = cos, 1t 1. 4 Find P(T = 0); the interquartile range. (Total 7 marks) 36. A continuous random variable X has probability density function defined by c 4 x 0,, for x 3 otherwise. 3 Find the exact value of the constant c in terms of. Sketch the graph of f (x) and hence state the mode of the distribution. Find the exact value of E(X). 8

9 (Total 1 marks) 37. A continuous random variable X has probability density function, 1x 1 x for 0 x1, 0, otherwise. Find the probability that X lies between the mean and the mode. GDC required 38. A company buys 44% of its stock of bolts from manufacturer A and the rest from manufacturer B. The diameters of the bolts produced by each manufacturer follow a normal distribution with a standard deviation of 0.16 mm. The mean diameter of the bolts produced by manufacturer A is 1.56 mm. 4.% of the bolts produced by manufacturer B have a diameter less than 1.5 mm. Find the mean diameter of the bolts produced by manufacturer B. A bolt is chosen at random from the company s stock. (d) Show that the probability that the diameter is less than 1.5 mm is 0.31, to three significant figures. The diameter of the bolt is found to be less than 1.5 mm. Find the probability that the bolt was produced by manufacturer B. Manufacturer B makes 8000 bolts in one day. It makes a profit of $1.50 on each bolt sold, on condition that its diameter measures between 1.5 mm and 1.83 mm. Bolts whose diameters measure less than 1.5 mm must be discarded at a loss of $0.85 per bolt. Bolts whose diameters measure over 1.83 mm are sold at a reduced profit of $0.50 per bolt. Find the expected profit for manufacturer B. (6) (Total 16 marks) 39. A random variable X is normally distributed with mean and variance. If P (X 6.) = and P (X 9.8) = , calculate the value of and of. 40. A business man spends X hours on the telephone during the day. The probability density function of X is given by 1 3 (8x x ), 1 0, for 0 x otherwise. (i) Write down an integral whose value is E(X). 9

10 (ii) Hence evaluate E(X). (i) Show that the median, m, of X satisfies the equation m 4 16m + 4 = 0. (ii) Hence evaluate m. Evaluate the mode of X. (Total 11 marks) 41. The weights in grams of bread loaves sold at a supermarket are normally distributed with mean 00 g. The weights of 88 of the loaves are less than 0 g. Find the standard deviation. 4. The time, T minutes, required by candidates to answer a question in a mathematics examination has probability density function 1 (1t t 7 f ( t) 0, 0), for 4 t 10 Find (i), the expected value of T; (ii), the variance of T. (7) A candidate is chosen at random. Find the probability that the time taken by this candidate to answer the question lies in the interval [, ]. (Total 1 marks) 43. A certain type of vegetable has a weight which follows a normal distribution with mean 450 grams and a standard deviation 50 grams. In a load of 000 of these vegetables, calculate the expected number with a weight greater than 55 grams. Find the upper quartile of the distribution. 44. The continuous random variable X has probability density function x 1 x 0,, for 0 x k otherwise. Find the exact value of k. 10

11 Find the mode of X. Calculate P(1 X ). (Total 10 marks) 45. The lengths of a particular species of lizard are normally distributed with a mean length of 50 cm and a standard deviation of 4 cm. A lizard is chosen at random. Find the probability that its length is greater than 45 cm. Given that its length is greater than 45 cm, find the probability that its length is greater than 55 cm. 46. The time, T minutes, spent each day by students in Amy s school sending text messages may be modelled by a normal distribution. 30 of the students spend less than 10 minutes per day. 35 spend more than 15 minutes per day. Find the mean and standard deviation of T. (6) The number of text messages received by Amy during a fixed time interval may be modelled by a Poisson distribution with a mean of 6 messages per hour. Find the probability that Amy will receive exactly 8 messages between 16:00 and 18:00 on a random day. Given that Amy has received at least 10 messages between 16:00 and 18:00 on a random day, find the probability that she received 13 messages during that time. (d) During a 5-day week, find the probability that there are exactly 3 days when Amy receives no messages between 17:45 and 18:00. (Total 18 marks) 47. The times taken for buses travelling between two towns are normally distributed with a mean of 35 minutes and standard deviation of 7 minutes. Find the probability that a randomly chosen bus completes the journey in less than 40 minutes. 90 of buses complete the journey in less than t minutes. Find the value of t. A random sample of 10 buses had their travel time between the two towns recorded. Find the probability that exactly 6 of these buses complete the journey in less than 40 minutes. (Total 11 marks) 48. A furniture manufacturer makes tables. A table leg is considered to be oversize if its width is 11

12 greater than 10.5 cm and undersize if its width is less than 9.5 cm. From past experience it is found that of the table legs that are made are oversize and that 4 of the table legs are undersize. The widths of the table legs are normally distributed with mean cm and standard deviation cm. Find the value of and of. 49. Juan plays a quiz game. The scores he achieves on the separate topics may be modelled by independent normal distributions. On the topic of sport, the scores have the distribution N (75, 1 ). Find the probability that Juan scores less than 57 points on the topic of sport. On the topic of literature, Juan s scores have a mean of 45, and 30 of his scores are greater than 50. Find the standard deviation of his scores on the topic of literature. Juan claims that he scores better in current affairs than in sport. He achieves the following scores on current affairs in 10 separate quizzes Perform a hypothesis test at the 5 significance level to decide whether there is evidence to support his claim. (6) (Total 11 marks) 50. A company produces computer microchips, which have a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of 3.7 months. If a microchip is guaranteed for 84 months find the probability that it will fail before the guarantee ends. The probability that a microchip does not fail before the end of the guarantee is required to be 99. For how many months should it be guaranteed? A rival company produces microchips where the probablity that they will fail after 84 months is Given that the life expectancy also follows a normal distribution with standard deviation 3.7 months, find the mean. 51. The distance travelled by students to attend Gauss College is modelled by a normal distribution with mean 6 km and standard deviation 1.5 km. (i) Find the probability that the distance travelled to Gauss College by a randomly selected student is between 4.8 km and 7.5 km. (ii) 15 of students travel less than d km to attend Gauss College. Find the value of d. (7) At Euler College, the distance travelled by students to attend their school is modelled by a normal distribution with mean km and standard deviation km. 1

13 If 10 of students travel more than 8 km and 5 of students travel less than km, find the value of and of. (6) The number of telephone calls, T, received by Euler College each minute can be modelled by a Poisson distribution with a mean of 3.5. (i) Find the probability that at least three telephone calls are received by Euler College in each of two successive one-minute intervals. (ii) Find the probability that Euler College receives 15 telephone calls during a randomly selected five-minute interval. (8) (Total 1 marks) 13