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QUESTION 6 If the probability of success in a Bernoulli trial is 0.30 the variance is A) 0.70 (B) 0.46 (C) 0.30 (D) 0.21
QUESTION 8 A test includes six multiple choice questions. Each question has four options for the answer. If the answers are guessed the probability of getting at most two questions correct is represented by 6 0 6 6 1 5 (A) 5 0.25 x 0.75° + 1 0.25 x 0.75 6 0 6 [0 1 s (6 2 4 (B) N 0.25 x 0.75° + 1 0.25 x 0.75° + 5 0.25“x 0.75 ©) 1 6 0.25% % 0.75° + (Jo.zsl x 0.755] @ oo 6 6 0.25% % 0.75° + (Jo.zsl x 0.75° + [2]0.252 x 0.754J
QUESTION 7 The records of a shoe manufacturer show that 10% of shoes made are defective. Assuming independence the probability of getting 2 defective shoes in a batch of 20 is A) 0.1937 (B) 0.2852 (C) 0.3917 (D) 0.6083
QUESTION 11 (4 marks) A sugar company samples the packets of sugar it produces and finds that 5% of packets are underweight. Consider a batch of 20 packets. a) Determine how many packets of sugar the company can expect to be underweight in a batch of 20 packets. [1 mark] b) Determine the variance of the batch. [1 mark] c) Determine the probability that at most one of the packets of sugar in the batch is underweight. [2 marks]
QUESTION 6 A random variable X 1s the number of successes in a Bernoulli experiment with #z trials each with a probability of success p and a probability of failure g. The probability distribution table of X is shown.
QUESTION 9 A basket contains 10 green apples and 30 red apples. Three apples are drawn at random from the basket with replacement. Determine the probability that exactly two of the three apples are green. 3 (A) o 9 (B) o 10 ©) o (D) 27 64
QUESTION 17 (3 marks) In any five-day working week Leonardo either catches a bus to work or uses another form of transportation. On average he catches the bus to work on three of the five days. His decision on any given day is independent of his decision on any other day. Determine the probability that Leonardo catches a bus to work on exactly one day in a given five-day working week.
QUESTION 6 When seeds of a certain variety of flower are planted the probability of each seed germinating is 0.8. If eight seeds are planted what is the probability that at least six seeds will germinate? A) 0.797 (B) 0.503 (C) 0.294 (D) 0.001
QUESTION 2 A binomial random variable arises from the number of successes in n independent Bernoulli trials. A context not suitable for modelling using a binomial random variable is recording the number of (A) heads when a coin is tossed 12 times. (B) left-handed people in a sample of 100 people. (C) times a player hits a target from 20 shots where each shot is independent of all other shots. (D) red marbles selected when three marbles are drawn without replacement from a bag containing four blue and five red marbles.
QUESTION 12 (3 marks) The probability that a debating team wins a debate can be modelled as a Bernoulli distribution. Given that the probability of winning a debate is 3 a) Determine the mean of this distribution. [1 mark] b) Determine the variance of this distribution. [1 mark] c¢) Determine the standard deviation of this distribution. [1 mark]
QUESTION 11 (7 marks) A salesperson has a 20% probability of making a sale to each customer who enters the store. Each sale is independent of all other sales. a) Determine the mean number of sales on a day where 25 customers enter the store. [2 marks] b) Determine the standard deviation of the number of sales on a day where 25 customers enter the store. [2 marks] c) Determine the minimum number of customers who would have to enter the store to have an 88% chance or more of making at least one sale. [3 marks]
QUESTION 18 (3 marks) The intelligence quotient (IQ) of individuals in a population is normally distributed with a mean of 100 and a standard deviation of 16. Nine individuals are chosen at random from the population. Determine the probability that no more than two of the individuals have an 1Q of at least 120.
QUESTION 1 The random variable X is binomially distributed with 10 trials and a probability of success equal to 0.25 at each attempt. The value of P(X = 1) is equal to @ () ® () © 1-() ™ 1-()
QUESTION 12 (5 marks) People with type O negative blood are said to be ‘universal donors`. In Australia 9% of the population has this blood type. On a given day a random group of 45 people volunteer to donate blood. a) Identify why this context is suitable for modelling as a binomial distribution. [1 mark] b) Determine the mean and standard deviation of the number of people who are universal donors. [2 marks] c) Determine the probability that no more than 3 of the donors are universal donors. [2 marks]
QUESTION 15 (8 marks) Suppose the proportion of Australians who supported the removal of single-use plastic bags from supermarkets is 64%. a) Using the normal approximation determine the probability that in a randomly selected sample of size 100 more than 70% of those surveyed supported the removal of the single-use plastic bags. [2 marks] b) Determine the size of the sample required for the survey to achieve a margin of error of 4% 1in an approximate 95% confidence interval for this proportion. [2 marks] c) Identify the effect that halving the margin of error has on the sample size obtained in 15b). [2 marks] d) Determine the probability that in a randomly selected sample of size 25 the sample proportion is equal to the population proportion. [2 marks]
QUESTION 3 A bag contains 10 buttons of the same shape and size in different colours: 5 blue 3 green and 2 red. If 3 buttons are randomly drawn from the bag which probability can be calculated using the binomial distribution? (A) P(3 green) with replacement (B) P(3 blue) without replacement (C) P(2 green and | red) with replacement (D) P(2 red and | blue) without replacement
QUESTION 11 (4 marks) Two random samples (A and B) were obtained using two different Bernoulli experiments. Each Bernoulli trial in the random samples was recorded as 1 (for success) or 0 (for failure). The results are shown. In sample A for each trial the mean is 0.8 and the variance is 0.16. a) Use the sample B results to determine the mean and variance for each trial in sample B. = /2 marks/ b) Compare the variability about the means of samples A and B. [2 marks]
QUESTION 13 (5 marks) At a certain airport the departure of one in five international flights is delayed every day. The status of any flight is independent of other flights. One international flight is selected at random each day for three days. Each selection is recorded as either ‘delayed’ or ‘not delayed’. a) State two conditions that make this context suitable for modelling using a binomial random variable. [2 marks] b) Calculate the probability that at least two of the selected flights were delayed. [3 marks]
QUESTION 15 (4 marks) In a certain game players throw one water balloon at a target. There is a one in four chance of hitting the target. a) State the probabilities of all the possible outcomes for one throw at the target. [2 marks] b) Let H be the discrete random variable for one of the possible outcomes. Determine the mean and variance of the distribution of random variable H when 20 players throw [2 marks] a water balloon at the target.
QUESTION 2 The probability of hitting a bullseye on a standard dartboard is 1 in 1250. What is the probability of hitting a bullseye exactly once in 10 attempts? (Das) (is) ()(cea) *(ias0) © (aa) *(i) (saa) *(ias0) (B) (D)
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