2 marks. b. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.

Transcription

1 2016 MATHMETH EXAM 2 16 Question 3 (16 marks) A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges. On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student s plugging-in is independent of any other student s correctness. a. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places. b. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley. Given this, find the probability that fewer than five laptops are not correctly plugged in. Give your answer correct to four decimal places. SECTION B Question 3 continued

2 MATHMETH EXAM 2 The time for which a laptop will work without recharging (the battery life) is normally distributed, with a mean of three hours and 10 minutes and standard deviation of six minutes. Suppose that the laptops remain out of the recharging trolley for three hours. c. For any one laptop, find the probability that it will stop working by the end of these three hours. Give your answer correct to four decimal places. A supplier of laptops decides to take a sample of 100 new laptops from a number of different schools. For samples of size 100 from the population of laptops with a mean battery life of three hours and 10 minutes and standard deviation of six minutes, P is the random variable of the distribution of sample proportions of laptops with a battery life of less than three hours. d. Find the probability that Pr(P 0.06 P 0.05). Give your answer correct to three decimal places. Do not use a normal approximation. 3 marks It is known that when laptops have been used regularly in a school for six months, their battery life is still normally distributed but the mean battery life drops to three hours. It is also known that only 12% of such laptops work for more than three hours and 10 minutes. e. Find the standard deviation for the normal distribution that applies to the battery life of laptops that have been used regularly in a school for six months, correct to four decimal places. SECTION B Question 3 continued TURN OVER

3 2016 MATHMETH EXAM 2 18 The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours. f. Suppose the supplier tests the battery life of the laptops one at a time. Find the probability that the first laptop found to have a battery life of less than three hours is the third one. 1 mark The laptop supplier finds that, in a particular sample of 100 laptops, six of them have a battery life of less than three hours. g. Determine the 95% confidence interval for the supplier s estimate of the proportion of interest. Give values correct to two decimal places. 1 mark SECTION B Question 3 continued

4 MATHMETH EXAM 2 h. The supplier also provides laptops to businesses. The probability density function for battery life, x (in minutes), of a laptop after six months of use in a business is ( 210 xe ) f( x) = x x 210 elsewhere i. Find the mean battery life, in minutes, of a laptop with six months of business use, correct to two decimal places. 1 mark ii. Find the median battery life, in minutes, of a laptop with six months of business use, correct to two decimal places. SECTION B continued TURN OVER

5 2017 MATHMETH EXAM 2 18 Question 2 (14 marks) Susie is gathering data on two particular species of ants, jumping jack ant and red fire ant. They are very difficult to tell apart and both species are equally likely to be caught. a. Let X be the random variable with values equal to the distance, in metres, of the jumping jack ant from an old log. The probability density function of X is 2x 0 x a 2 f( x) a 0 otherwise i. It is known that the mean distance of a jumping jack ant from the old log is 120 metres. Show that the value of a is 180. ii. Find the probability that a jumping jack ant is more than 150 metres from the old log. Copyright Insight Publications 2017 SECTION B Question 2 continued

6 MATHMETH EXAM 2 One technique for distinguishing between the two types of ant is to measure the length of their bodies. For red fire ants it is known that their body lengths are normally distributed with a mean of 25 mm and a standard deviation of 5 mm. b. Find the probability that a randomly chosen red fire ant has a body length that is shorter than 18 mm. Give your answer correct to four decimal places. c. It is also known that 10% of jumping jack ants have a body length shorter than 20 mm and 10% of jumping jack ants have a body length longer than 28 mm. Assuming that the body length of a jumping jack ant is normally distributed, find the mean and the standard deviation of the body length of a jumping jack ant. Give your answer correct to two decimal places. 3 marks Copyright Insight Publications 2017 SECTION B Question 2 continued TURN OVER

7 2017 MATHMETH EXAM 2 20 d. During her studies, Susie finds a particular site near the coast that has these two types of ants in abundance. 70% of them are jumping jack ants and 30% are red fire ants. i. Susie examines a single ant from this site and finds its body to be shorter than 18 mm. What is the probability that it is a jumping jack ant? Give your answer correct to three decimal places. 3 marks ii. Susie selects a sample of 500 ants from the site. Let ˆP be the proportion of jumping jack ants in the sample. Use normal approximation to find the probability that the proportion of jumping jack ants in the sample is greater than Give your answer correct to four decimal places. Copyright Insight Publications 2017 SECTION B continued

8 VCE Mathematical Methods Units 3&4 Trial Examination 2 Question and Answer Booklet Question 3 (19 marks) A school has a total of 1500 students of which 150 are known to ride to school everyday. The school takes a number of independent random samples of its students. Each sample contains 20 students. Let X be the random variable that represents the number of students in a sample that ride to school. a. i. Find Pr( X < 3). Give your answer correct to four decimal places. ii. Given that less than 3 students in a sample ride to school, find the probability that none of the students in the sample ride to school. Give your answer correct to four decimal places. b. Find the smallest sample size that would need to be used to ensure the probability at least one student in the sample rides to school is at least 80%. 14 MMU34EX2_QA_2017.FM Copyright 2017 Neap

9 VCE Mathematical Methods Units 3&4 Trial Examination 2 Question and Answer Booklet Let Pˆ be the random variable of the distribution of sample proportions of students who ride to school. c. i. Find the expected value of Pˆ. 1 mark ii. Find the standard deviation of Pˆ with a sample size of 20. Give your answer correct to four decimal places. 1 mark iii. Find the new sample size required to halve the standard deviation of Pˆ. 1 mark For those students that do ride to school, the distance that each student rides is known to be normally distributed with a mean of 2 km and a standard deviation of 400 m. d. i. Find the probability that a particular student who rides to school rides between 1.5 km and 2 km. Give your answer correct to four decimal places. ii. Hence, find the probability that a student rides at least 2 km, given that they ride at least 1.5 km. Give your answer to four decimal places. 3 marks Copyright 2017 Neap MMU34EX2_QA_2017.FM 15

10 VCE Mathematical Methods Units 3&4 Trial Examination 2 Question and Answer Booklet The school embarks on a campaign to increase the number of students who ride to school each day. At the end of the campaign they survey 100 students and find that 20 students ride to school. e. An approximate 95% confidence interval for the population proportion that corresponds a b to this sample proportion is found to be -----, where a and b are integers. Find a and b. Use an integer multiple of the standard deviation in your calculations. f. Of the 20 students in the sample that ride to school, 12 ride further than 2 km and 6 ride between 1.5 km and 2 km, and the distance they ride is normally distributed. Find the new mean and standard deviation for the distance these students ride to school. Give your answer correct to four decimal places. 3 marks 16 MMU34EX2_QA_2017.FM Copyright 2017 Neap

11 Mathematical Methods Trial Examination Section B Page 18 Question 3 (21 marks) Car tyres come in many different brands. Even on a car, the tyres can wear out after travelling different distances, due to cornering, braking and wheel alignment. Manufacturers of car tyres are interested in the average distance in kilometres travelled by tyres, or the lifetime of tyres until they are considered un-usuable and need to be replaced. At this time a tyre is considered un-roadworthy. a. For brand A of car tyres, records show that the lifetime is normally distributed with a mean of 45,000 km and a standard deviation of 8,000 km. i. Find the probability that a brand A tyre will last for at least 40,000 km. Give your answer correct to four decimal places. 1 mark ii. For a random sample of 5 tyres of brand A, find the probability more than half will last longer than 40,000 km. Give your answer correct to four decimal places. 1 mark b. Brand B of car tyres, have a lifetime which is normally distributed. Records show that 11% of these tyres have a lifetime of more than 61,000 km while 20% become un-usuable before travelling 42,500 km. Determine the mean and standard deviation of the lifetime of brand B, giving your answers to the nearest thousand km. 3 marks Kilbaha Multimedia Publishing This page must be counted in surveys by Copyright Agency Limited (CAL)

12 Mathematical Methods Trial Examination Section B Page 19 c. Consider another brand C of car tyres. From a random sample of 36 car tyres, it has been found that 26 of these car tyres last longer than 40,000 km. Find a 95% confidence interval corresponding to the sample proportion of brand C car tyres which last longer than 40,000 km. Give values correct to three decimal places. 1 mark d. Consider another brand D of car tyres. It has been found that the probability that these car tyres which last longer than 40,000 km is 0.7. For samples of 36 tyres of brand D, ˆP is the random variable of the sample proportion of these tyres which last longer than 40,000 km. i. Find the expected value and variance of ˆP ii. Find the probability that the sample proportion of these car tyres lies within two standard deviations of the mean. Give your answer correct to three decimal places. Do not use a normal approximation. Kilbaha Multimedia Publishing This page must be counted in surveys by Copyright Agency Limited (CAL)

13 Mathematical Methods Trial Examination Section B Page 20 e. The lifetime T in thousands of km of brand E of car tyres, satisfies a probability density function given by 2 at 0 t 50 f t b 100 t 50 t elsewhere 3 3 i. Show that a and b. 312,500 6,250 ii. For 36 tyres of brand E, find the expected number that will last longer than 40,000 km. 1 mark Kilbaha Multimedia Publishing This page must be counted in surveys by Copyright Agency Limited (CAL)

14 Mathematical Methods Trial Examination Section B Page 21 iii. Sketch the graph of T on the axes below, clearly labelling the scale. iv. Find the expected value and variance of brand E car tyres. Kilbaha Multimedia Publishing This page must be counted in surveys by Copyright Agency Limited (CAL)

15 Mathematical Methods Trial Examination Section B Page 22 v. Find the probability that the lifetime of brand E car tyres lies within two standard deviations of the mean. Give your answer correct to three decimal places. vi. Find the median lifetime of brand E car tyres. Give your answer correct to three decimal places. Kilbaha Multimedia Publishing This page must be counted in surveys by Copyright Agency Limited (CAL)

16 2016 MATHMETH EXAM 2 14 Question 3 (18 marks) Tom is a keen gardener and owns a garden nursery. He specialises in tomato plants and grows two varieties cherry tomatoes and roma tomatoes. The height in centimetres of the cherry tomato plants that Tom is selling is normally distributed with a mean of 15 cm and a standard deviation of 4 cm. There are 4000 cherry tomato plants in his nursery. a. Tom classifies the tallest 15% of his cherry tomato plants as advanced. What is the minimum height of an advanced tomato plant, correct to the nearest millimetre? 1 mark Tom thinks that some of his cherry tomato plants are not growing quickly enough and decides to move them to a special greenhouse. He will move the cherry tomato plants that are less than 9 cm in height. b. How many cherry tomato plants will Tom move to the greenhouse? Give your answer correct to the nearest whole number. Copyright Insight Publications 2016 SECTION B Question 3 continued

17 MATHMETH EXAM 2 The height, x centimetres, of the roma tomatoes in the nursery follows the probability distribution density function where ( x 6) ksin, 6 x 16 hx ( ) 10 0, otherwise c. Show that k. 20 d. State the mean height of a roma tomato plant at the nursery. 1 mark Copyright Insight Publications 2016 SECTION B Question 3 continued TURN OVER

18 2016 MATHMETH EXAM 2 16 Tom thinks that the smallest 10% of the roma plants should be given a fertilizer. e. Find the maximum height correct to the nearest millimetre of a roma tomato plant that should be given a fertilizer. Tom classifies his roma tomato plants as either standard or tall. He knows that 20% of his roma tomato plants are tall. A customer, Katie, selects five roma tomato plants. She chooses each plant individually and finds that her decision on which plant to choose depends on her previous choice. If she chooses a tall plant then she has a probability of 1 4 of choosing a tall plant as her next plant. If she chooses a standard plant then she has a probability of 2 3 of choosing a standard plant as her next plant. The first one she chooses is tall. f. What is the probability that of the next four plants he chooses exactly two tall plants? Copyright Insight Publications 2016 SECTION B Question 3 continued

19 MATHMETH EXAM 2 Tom also has parsley plants in his nursery. He obtains his parsley plants from growers all over Australia. It is known that 30% will be prone to a particular leaf disease. Tom decides to test his plants for the leaf disease. He takes a random sample of 20 parsley plants. g. i. What is the probability that the sample proportion is equal to the population proportion of 0.3? Give your answer correct to four decimal places. Do not use a normal approximation. ii. What is the probability that the sample proportion lies within two standard deviations of the population proportion? Give your answer correct to four decimal places. Do not use a normal approximation. 3 marks Copyright Insight Publications 2016 SECTION B Question 3 continued TURN OVER

20 2016 MATHMETH EXAM 2 18 While testing for the leaf disease, Tom sees that some of his plants are infested with a leaf moth. He finds that 236 plants out of 500 are infested with the leaf moth. h. Find an approximate 95% confidence interval for the proportion of plants infested with the leaf moth. Give your answer correct to three decimal places. 1 mark Another 26 nursery owners from around Australia independently sample parsley plants from their stocks and test for leaf disease. Each calculates an approximate 95% confidence interval for p, the proportion of plants in the population infested with the leaf moth. It is subsequently found that of these 27 confidence intervals exactly one does not contain the value of p. Researchers investigating the prevalence of leaf moth randomly select five of the confidence intervals calculated by the nursery owners. i. What is the probability that exactly three of the selected confidence intervals contain the value of p? Give your answer correct to four decimal places. Copyright Insight Publications 2016 SECTION B continued

21 Question 3 A bank has over one million customers. The proportion of these customers who don t use online banking is 2 9. The bank takes a number of random samples of its customers. Each sample contains 15 customers. Let X be the random variable that represents the number of customers in a sample who don t use online banking. a. Find Pr(X 5). Give your answer correct to three decimal places. Let Pˆ be the random variable of the distribution of sample proportions of customers who don t use online banking. b. Find i. the expected value of Pˆ. 1 mark ii. the standard deviation of Pˆ. c. Suppose the bank wanted a standard deviation of Pˆ of less than 0.1. What is the minimum number of customers that the bank would need to have in each sample in order to achieve this?

22 d. The bank decides to retain its original sample size of 15 customers. Find the probability that a sample proportion would lie within one standard deviation of the population proportion. Do not use a normal approximation. Give your answer correct to three decimal places. 3 marks The duration of calls, in minutes, made by customers to the bank s telephone banking service is a continuous random variable T with a probability density function f, given by 3t (20 t), f ( t) , 2 t 15 elsewhere e. i. Find the mean duration of a call. Give your answer in minutes, correct to three decimal places. ii. Find the probability that a call took more than ten minutes. Give your answer correct to three decimal places.

23 iii. Find the probability that a call took more than twelve minutes given that it took more than ten minutes. Give your answer correct to three decimal places. The bank sampled 2000 of its customers across its entire customer base. It was found that 1860 of these customers were satisfied with their last interaction with the bank. f. Use this sample to find an approximate 90% confidence interval for the proportion of the population of the bank s customers who were satisfied with their last interaction with the bank. Give values correct to three decimal places. 1 mark

24 MATHEMATICS METHODS 6 Question 12 CALCULATOR-ASSUMED SAMPLE EXAMINATION (1) Rebecca sells potatoes at her organic fruit and vegetable shop that have weights normally distributed with a mean of 230 g and a standard deviation of 5 g. (a) Determine the probability that one of Rebecca s potatoes, selected at random, will weigh between 223 g and 235 g. (1 mark) (b) (c) Five percent of Rebeca s potatoes weigh less than w g. Determine w to the nearest gram. () A customer buys twelve potatoes. (i) (ii) Determine the probability that all twelve potatoes weigh between 223 g and 235 g. () If the customer is selecting the twelve potatoes one at a time, determine the probability that it takes the selection of eight potatoes before six potatoes weighing between 223 and 235 g have been selected. DO NOT WRITE IN THIS AREA AS IT WILL BE CUT OFF ( 3 marks) See next page

25 CALCULATOR-ASSUMED 7 SAMPLE EXAMINATION MATHEMATICS METHODS Rebecca also sells oranges. The weights of these oranges are also normally distributed. It is known that 5% of the oranges weigh less than 153 g while 12% of the oranges weigh more than 210 g. (d) Determine the mean and standard deviation of the weights of the oranges. (4 marks) DO NOT WRITE IN THIS AREA AS IT WILL BE CUT OFF See next page

26 CALCULATOR-ASSUMED 13 SAMPLE EXAMINATION Question 17 MATHEMATICS METHODS (8 marks) A random sample of 100 people indicated that 19% had taken a plane flight in the last year. (a) Determine a 90% confidence interval for the proportion of the population that had taken a plane flight in the last year. () DO NOT WRITE IN THIS AREA AS IT WILL BE CUT OFF Assume the 19% sample proportion applies to the whole population. (b) (c) A new sample of 100 people was taken and X = the number of people who had taken a plane flight in the last year was recorded. Give a range, using the 90% confidence interval, within which you would expect X to lie. (1 mark) Determine the probability that in a random sample of 120 people, the number who had taken a plane flight in the last year was greater than 26. (3 marks) See next page

27 MATHEMATICS METHODS 14 CALCULATOR-ASSUMED SAMPLE EXAMINATION Question 17 (continued) (d) If seven surveys were taken and for each a 95% confidence interval for p was calculated, determine the probability that at least four of the intervals included the true value of p. () DO NOT WRITE IN THIS AREA AS IT WILL BE CUT OFF See next page

28 CALCULATOR-ASSUMED 15 SAMPLE EXAMINATION Question 18 MATHEMATICS METHODS (10 marks) A random survey was conducted to estimate the proportion of mobile phone users who favoured smart phones over standard phones. It was found that 283 out of 412 people surveyed preferred a smart phone. (a) Determine the sample proportion p of those in the survey who preferred a smart phone. (1 mark) DO NOT WRITE IN THIS AREA AS IT WILL BE CUT OFF (b) (c) Use the survey results to estimate the standard deviation of p, for the sample proportions. () A follow-up survey is to be conducted to confirm the results of the initial survey. Working with a confidence interval of 95%, estimate the sample size necessary to ensure margin of error of at most 4%. (3 marks) The 90% confidence interval of the sample proportion p from the initial survey is p (d) Use the 90% confidence interval of the initial sample to compare the following samples: (i) A random sample of 365 people at a shopping centre found that 258 had a preference for a smart phone. () (ii) A random sample of 78 people at a retirement village found that 32 had a preference for a smart phone. () See next page

29 CALCULATOR-ASSUMED 19 SAMPLE EXAMINATION Question 21 MATHEMATICS METHODS (5 marks) The graph below shows the number of faulty batteries per packet of 50 AAA batteries, when 50 packets are sampled at random. DO NOT WRITE IN THIS AREA AS IT WILL BE CUT OFF (a) Identify the type of distribution of X = the number of faulty batteries per packet of 50 AAA batteries. ( 1 mark) A manufacturer of AAA batteries assumes that 99% of the batteries produced are fault-free. Ten samples of 50 packets of 50 AAA batteries are selected at random and tested. The number of faulty batteries in each of the 10 random samples is shown below. Sample Number of faulty batteries (b) Using the assumption that 99% of batteries are fault free calculate the 95% confidence interval for the proportion of faulty batteries expected when sampling. (3 marks) (c) Decide which of the samples, if any lie outside the 95% confidence level. (1 mark) End See of next questions page