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QUESTION 4 Pulse rates of adult men are approximately normally distributed with a mean of 70 and a standard deviation of 8. Which of the following choices correctly describes how to determine the proportion of men that have a pulse rate greater than 787 (A) Determine the area to the left of z= 1 under the standard normal curve. (B) Determine the area to the right of z = 1 under the standard normal curve. (C) Determine the area to the right of z = —1 under the standard normal curve. (D) Determine the area between z =—1 and z = 1 under the standard normal curve.
QUESTION 13 (6 marks) An online retailer claims that 90% of all orders are shipped within 12 hours of being received. On a particular day 121 orders were received and 102 orders were shipped within 12 hours. a) State the sample proportion of orders shipped within 12 hours. [1 mark] The distribution of the sample proportion of all orders that are shipped within 12 hours of being received on any day is approximately normal. b) Assuming the online retailer`s claim is true find the probability that in a random sample of 121 less than 85% of all orders are shipped within 12 hours. [3 marks] c) Use the result from 13b) to evaluate the reasonableness of the online retailer`s claim. [2 marks]
QUESTION 16 (4 marks) Bottles of soft drink should contain a volume with a mean of 591 mL but some variation is expected. Any bottle at or below the 20th percentile of the volume distribution is rejected. A percentile is a measure in statistics that shows the values below which a given percentage of observations occur. Thirty-five per cent of the bottles contain 593 mL or more of soft drink. Assuming the volumes are normally distributed determine the smallest volume (in mL) that will be accepted.
QUESTION 1 The scores obtained on a test can be assumed to be normally distributed with a mean of 102 and a standard deviation of 19. What proportion of scores are over 113? (A) 0.2813 (B) 0.5789 (C) 0.7187 (D) 0.8216
QUESTION 14 (7 marks) The heights of students at School A are normally distributed with a mean of 165 cm and a standard deviation of 15 cm. a) Determine the probability that a student chosen at random from School A is shorter than 180 cm. [1 mark] b) Determine the minimum integer value of the height of a student who is in the top 2% of this distribution. [3 marks] The heights of students at School B are also normally distributed. A student at School B has the same height as the height determined in Question 14b) but their corresponding z-score is 3. c) Determine which student`s height ranks higher in terms of percentile for their school. [3 marks]
QUESTION 19 (4 marks) A random variable X defined over the interval a < x < b is uniformly distributed if its probability density function is defined by: I —— a<x<bh f(x)=1(b=a)` 0 otherwise The expected value and variance of a uniform random variable X are 2 E(X):@ Var(X)= (b;;) A manufacturer has observed that the time that elapses between placing an order with a supplier and the delivery of the order is uniformly distributed between 100 and 180 minutes. Determine the probability that the time between placing an order and delivery of the order will be within one standard deviation of the expected time.
QUESTION 20 (3 marks) The random variable B is normally distributed with a mean of 0 and a standard deviation of 1. Determine the probability that the quadratic equation x? +3x+2B =0 has real roots.
QUESTION 4 The weekly amount of money a company spends on repairs is normally distributed with a mean of $1200 and a standard deviation of $100. Given that P(Z <-2.5) =0.0062 and P(Z >1) = 0.1587 where Z is a standard normal random variable determine the probability that the weekly repair costs will be between $950 and $1300. (A) 0.6525 (B) 0.6587 (C) 0.8351 (D) 0.8413
QUESTION 5 Which normal distribution curve best represents a normal distribution with a mean of 1 and a standard deviation of 0.5?
QUESTION 18 (4 marks) A percentile 1s a measure in statistics showing the value below which a given percentage of observations occur. The continuous random variable X has the probability density function 2x—2 1<x<2 f(x)={ 0 otherwise Determine the 36th percentile of X.
QUESTION 6 A stall at the school fete sells cups of lemonade. Assuming the amount of lemonade in a cup 1s normally distributed with a mean of 60 mL and a standard deviation of 3 mL 80% of the cups contain more than (A) 52.4mL (B) 57.5mL (C) 61.6mL (D) 62.5mL
QUESTION 12 (4 marks) Suppose that the distance travelled by vehicles in a year can be modelled by a normal distribution. In 2021 vehicles travelled a mean of 13 700 km with a standard deviation of 3400 km. a) Determine the probability that a vehicle chosen at random travelled less than 12 000 km in 2021. [2 marks] b) Determine the value of x where 60% of vehicles travelled less than x km in 2021. [2 marks]
QUESTION 14 (8 marks) Ravi randomly sampled 200 different pet owners in Brisbane and found that 50 celebrate their pet`s birthday. a) Determine an approximate 95% confidence interval for the proportion of Brisbane pet owners who celebrate their pet`s birthday. [2 marks] Two of Ravi`s friends also randomly sampled Brisbane pet owners. The results are shown in the table. Friend`s name | Number sampled | Number who celebrate their pet`s birthday Khadija suggested a more precise estimate for the proportion of Brisbane pet owners who celebrate their pet`s birthday could be obtained by combining their results. b) Using all available data determine an approximate 95% confidence interval for the proportion of Brisbane pet owners who celebrate their pet`s birthday. [2 marks] c) Use the results from Questions 14a) and 14b) to evaluate the reasonableness of Khadija`s suggestion. [2 marks] The proportion of all Brisbane pet owners who celebrate their pet`s birthday is 0.24. d) Using the normal approximation determine the probability that in a randomly selected sample of size 200 more than 30% of pet owners celebrate their pet`s birthday. [2 marks]
QUESTION 16 (4 marks) The time spent waiting in a queue at a certain supermarket is given by (X +1 1) minutes where X is a random variable with the probability density function al4-x* /()= (T) 0 otherwise —2€x£2 Determine the probability of waiting between 10 and 12 minutes in a queue at this supermarket.
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 9 The birth mass of babies is normally distributed with a mean 3500 grams and a standard deviation 500 grams. The probability that the birth mass of a baby is less than 3200 grams is (A) 0.01 (B) 0.06 (C) 0.22 (D) 0.27
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 19 (5 marks) Consider the following information when completing this question. If X is normally distributed with mean p and standard deviation o then X~N(u 0%) If X and Y are two independent random normal variables such that X~N(py 0 %) and Y~N (u 0 °) then X —Y~N(uy — phz 01° + 0 °) and X + Y `N(‘ul + flz 0-12 + 0-22) Contestant A and contestant B have trained to navigate an obstacle course. The times for each contestant to run the obstacle course are independent of each other. On any given day the time to run the obstacle course for each contestant is normally distributed. Let A be the run time (in minutes) for contestant A and B be the run time (in minutes) for contestant B with A~N(80 10%) B~N(78 122) Determine the probability that contestant B runs the obstacle course faster than contestant A.
QUESTION 17 (5 marks) Model bridges were constructed for a competition. The models that could support the heaviest loads before collapsing were given awards. The load results of the competition were normally distributed with a mean of 1.36 kg and a standard deviation of 0.12 kg. Three award categories were used: honours for the top 15% of load results; distinction for the next 15%; and commendation for the next 15%. The model bridge constructed by Finley only just missed out on a commendation. Kirby’s model bridge only just qualified for honours. Determine the difference to the nearest gram between the loads supported by Finley and Kirby’s models.
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