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QUESTION 3 According to a recent census the mean hours worked per week by all Australian workers is 35.6 hours. A mean of 36.1 hours worked per week is calculated from a random selection of 500 Australian workers. Based on this data which of the following is correct? (A) x=35.6 u=36.1 (B) X =35.6 X=36.1 (C) x=36.1 u=35.6 (D) x=36.1 X =356
QUESTION 3 The masses of packages of cheese produced by a company are assumed to be normally distributed with a known mean of ¢ grams and a standard deviation of 7.37 grams. The packages of cheese are labelled to contain 500 grams. Given there 1s a 25% probability that the mean mass of 20 randomly selected packages will be less than the labelled amount the value of u 1s (A) 498.89 (B) 500.25 (C) 50111 (D) 504.98
QUESTION 13 (6 marks) Data records show that the speeds of cars at a particular location on a highway are normally distributed with a mean of 98.7 km h™! and a standard deviation of 4.1 km h™. The speed limit at this location is 100 km h ™. A police officer plans to record the speeds of 20 randomly selected cars at this location. a) Determine the expected number of cars in the sample that will be travelling within + 1 km h™! of the population mean. [2 marks] b) Determine the probability that the mean speed of this sample will exceed the speed limit. [2 marks] There 1s a 5% probability that the mean speed of this sample will exceed £. c) Determine the value of 4. [2 marks]
QUESTION 1 Which of the following is a population parameter? (A) s B) u C) x (D) z
QUESTION 4 The number of sunflower seeds in each packet produced by a company is known to be normally distributed with a standard deviation of 100. A worker counts the number of seeds in a random sample of four packets and calculates the sample mean. Based on this sampling the standard deviation of the distribution of the sample mean is (A) 25 (B) 50 C) 75 (D) 100
QUESTION 10 The 2016 Australian census recorded the number of bedrooms per household. The results are summarised in the histogram as shown. Based on this data the mean number of bedrooms per household was calculated to be 3.5. Frequency 0 1 2 3 4 5 6 Number of bedrooms Fifty samples of size 40 were randomly selected from the census data and the sample means recorded. The histogram that most likely represents the distribution of the sample means is A 20 (B) 204 Frequency S Frequency S ] 0 0 1.5 2.0 25 3.0 3540 45 5.0 5.5 3.0 3.1 3.2 33 34 35 36 3.7 3.8 Number of bedrooms Number of bedrooms <€) 20 (D) 204 > 2 = = Q O =10 S 10 & o ~ L 0 0 3.1 3233343536237 3839 3.1 32 33 34 35 3.6 3.7 3.8 39 Number of bedrooms Number of bedrooms
QUESTION 4 The mean time that visitors spend at an art exhibition is 39 minutes and the standard deviation is 6 minutes. Determine the approximate probability that the mean time spent at the exhibition by a random sample of 35 visitors is between 38 and 40 minutes. (A) 0.13 (B) 0.16 (C) 0.68 (D) 0.84
QUESTION 12 (6 marks) The life span of batteries manufactured by a company is assumed to be normally distributed with an unknown mean and standard deviation. A supervisor at the company randomly selects 7 batteries and uses the life spans from this sample to calculate an approximate 95% confidence interval for the population mean of (2321.4 2423.6) hours. a) Determine the mean life span for this sample of batteries. [1 mark] The standard deviation of the life spans of batteries in this sample is 125 hours. b) Determine n. [3 marks] c) Use the result from Question 12b) to explain whether the assumption that the life span of batteries is normally distributed is required to support the supervisor’s calculations. [2 marks]
QUESTION 19 (7 marks) Consider the following information. Continuous random variable X variance The waiting time (minutes) until workers at a certain call centre receive their nth phone call where ne Z¥ is a random variable 7 with probability density function kn tn—l _% >0 f(t)= (n—l)!e t 0 otherwise where k 1s a positive constant. The waiting time until workers receive their 5th call is collected from a random sample of 80 workers. Determine the probability that the mean waiting time from this sample is more than 16 minutes.
QUESTION 2 Which statement regarding sample means is true? (A) The distribution of X is always normally distributed. (B) The distribution of X is always normally distributed. (C) The value of x changes when different samples are selected. (D) The value of u changes when different samples are selected.
QUESTION 9 A random variable X 1s normally distributed with a mean of 36 and a standard deviation of 4. The respective mean and standard deviation of the distribution of X from repeated random samples of size 9 are (A) 4 and 4 9 (B) 4and 4 3 (C) 36 and g (D) 36 and %
QUESTION 10 In a town the mean number of residents per household is 3.79 people with a standard deviation of 1.47 people. Using a random sample of 45 households from the town determine the probability that the mean number of residents per household will be more than 4. (A) 0.17 (B) 0.33 (C) 0.83 (D) 0.96
QUESTION 13 (5 marks) An article claims that the mean starting salary of graduates in Australia is currently $64 800 with a standard deviation of $4500. To check the validity of this claim an employment agent intends to collect data on the starting salaries of a random sample of 360 graduates. a) Determine the probability that the sample mean starting salary will be between $64 000 and $65 000. [2 marks] From the data the agent calculates a confidence interval for the population mean starting salary of (564 589 $65 811). b) Determine the sample mean. [1 mark] c) Comment on the reasonableness of the article’s claim based on this confidence interval. [2 marks]
QUESTION 2 The time taken for customers to assemble a particular piece of furniture is normally distributed with a mean of 4 hours and a standard deviation of 36 minutes. A random sample of 9 customers is selected. Each customer assembles the piece of furniture. The mean and standard deviation of the distribution of the mean time taken to assemble the piece of furniture for this sample is (A) g hours 4 minutes. (B) ghours 12 minutes. (C) 4 hours 4 minutes. (D) 4 hours 12 minutes.
QUESTION 17 (7 marks) At a honey bottling factory the honey dispensing machine is designed to fill each bottle with masses that are normally distributed with a mean of 503.5 grams and a standard deviation of 17.2 grams. The company randomly samples bottles of honey so that the probability of the mean mass being greater than 500 grams is 0.9. a) Determine the sample size that the company used. [5 marks] A worker at the factory claims that the results of the sample from 17a) would not be valid as the mass of honey dispensed into each bottle cannot be assumed to be normally distributed. b) Comment on the reasonableness of this claim. [2 marks]
QUESTION 10 A random variable is drawn from a population with the distribution shown in the histogram. Frequency 0 4 8 12 16 20 A number of samples of size 10 were randomly selected from this distribution and the sample means xX were recorded. The histogram that most likely represents the distribution of the sample means is (A) (B) 1 Frequency Frequency T TT 0 4 8 12 16 20 0 4 8 12 16 20 (C) (D) > > S S oO oO ; 3 3 3 fo fa 0 4 8 12 16 20 0 4 8 12 16 20
QUESTION 2 The standard deviation for the scores of 1000 students completing an entry test at a certain university is 13. A researcher takes repeated random samples of the test results with each sample comprising 40 scores and calculates the mean score for each sample. Determine the standard deviation of the distribution of the sample mean scores. (A) 3.08 (B) 2.06 (C) 0.41 (D) 0.33
QUESTION 15 (7 marks) The travel time for students attending a certain university is assumed to be normally distributed with a population mean of 25.2 minutes and standard deviation of 4.7 minutes. Travel times are collected from a random sample of 120 of these students and used to calculate a sample mean X | 1N minutes. a) Determine P(X i< 25). [2 marks] b) Given P(X 1> k) = (0.9 determine the value of k. [1 mark] Travel times are collected from a second random sample of the university’s students and used to calculate a second sample mean X’> in minutes. c) Given P(X < 25) 0.4 determine the number of students in the second sample. [4 marks] 2
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