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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 9 The scores on a test are assumed to be normally distributed. Researchers use the results from a random sample of scores to calculate a confidence interval for the population mean. However a shorter confidence interval width is required so the researchers decide to use a second sample for their calculations. Assuming that the standard deviations for both samples are the same the researchers can ensure that a shorter confidence interval width is produced by (A) decreasing the sample size and decreasing the confidence level. (B) decreasing the sample size and increasing the confidence level. (C) 1increasing the sample size and decreasing the confidence level. (D) increasing the sample size and increasing the confidence level.
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 7 The heights of all students at a school were measured. A mean height of 157.0 cm was calculated from this data. A random sample of 35 students from this school was selected. The mean height of this sample was 159.7 cm with a standard deviation of 8.7 cm. The smallest confidence level that could be used to produce a confidence interval that contains u« based on this sample 1s A) 85% (B) 90% C) 95% (D) 99%
QUESTION 10 The time taken by the Year 7 students at a particular school to complete a standardised test is known to be normally distributed. A researcher claims that the population mean is 8.2 minutes. The mean time taken to complete this test by a sample of 10 of these students is 8.1 minutes with a standard deviation of 1.2 minutes. The 95% confidence interval for u based on this sample is (A) (7.36 8.84) minutes (B) (7.46 8.94) minutes (C) (7.86 8.33) minutes (D) (7.96 8.44) minutes
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 18 (6 marks) The mass of a certain species of kangaroo is known to be normally distributed with a mean mass of u kg and standard deviation of ¢ kg. When one of the kangaroos is randomly selected the probability that its mass is greater than 83.2 kg is 0.145. When a sample of 12 kangaroos is randomly selected the probability that the sample mean mass is less than 74.1 kg is 0.079. A 90% approximate confidence interval for u is calculated using a random sample of n of the kangaroos that has a sample mean mass of 79.1 kg and a sample standard deviation equal to o. Determine the possible range of values that n could have been given that the confidence interval did not contain u.
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 7 The mass of a particular variety of cake is claimed to be normally distributed with a mean of 660 grames. A random sample of five of these cakes is found to have a mean mass of 600 grams. Which option represents an approximate confidence interval for z based on this sample? A B 600 660 600 660 grams grams ©) A (D) A 600 660 600 660
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 1 The time taken to complete orders at a pizza store is normally distributed with a mean time () of 10 minutes. The owner of the pizza store records the time taken to complete orders for a random sample of 20 pizzas each day over a 30-day period. From this data an approximate 90% confidence interval for 4 is calculated at the end of each day. How many of these confidence intervals would be expected to contain z? (A) 3 (B) 18 C) 27 (D) 30
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 10 A random variable is normally distributed with a mean #. An approximate 95% confidence interval for from a sample from this distribution is (209.7 221.9). An approximate confidence interval for 1 based on the same sample using a confidence level greater than 95% could be (A) (206.5 223.3) ®) ( ) (©) (210.6 221.0) D ( ) 208.5 223.1 215.8 228.0
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 5 Four random samples of different sizes were taken to estimate a certain population mean given a known population standard deviation. A 95% confidence interval was calculated for each sample. Sample 1 — Sample 2 | Sample 3 P Sample 4 P -~ T 1T 1T 71 > 10 15 20 25 30 35 40 Which sample used the largest sample size? (A) (B) ©) (D) Sample 1 Sample 2 Sample 3 Sample 4
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 5 A random sample of the petrol price per litre at 50 petrol stations produced a sample mean of $1.52 and a standard deviation of $0.14. Based on this sample and using a z-value of 1.5 an approximate confidence interval for u is (A) ($1.47 8%1.57) (B) ($1.48 $1.56) (C) ($1.49 $1.55) (D) ($1.50 $1.54)
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 17 (6 marks) The mass of a population of elephants is known to be normally distributed. A biologist randomly selects a number of elephants from this population and measures their masses. The mean mass of the sample is 5206 kg with a standard deviation of 356 kg. The biologist uses the data to calculate a 90% confidence interval for the population mean mass of (5159.1 5252.9) kg. Determine a 99% confidence interval for the population mean mass based on the same data. pop
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 5 A matchbox company advertises that the mean number of matches in their boxes is 50. Peter takes samples of the number of matches in 350 boxes. Based on his sample mean and corresponding sample standard deviation results he calculates a 95% confidence interval for the mean of (45.2 49.6) matches. Which of the following statements best describes a statistical inference from this confidence interval? (A) The company is using false advertising. (B) 95% of the company’s matchboxes do not contain 50 matches. (C) There is a statistical basis for being suspicious of the company’s advertising. (D) The sample size is too small to draw a conclusion about the company’s advertising.
QUESTION 7 Which of the following best describes an interval estimate of a population parameter? (A) the degree of uncertainty associated with an interval (B) the proportion of values expected to lie within an interval (C) aninterval that is calculated using the population parameter (D) an interval that is likely to include the unknown value of the population parameter
QUESTION 13 (5 marks) The heights of a sample of male and female students in a secondary school are summarised below. Mean height (cm) Standard deviation (cm) a) Determine the approximate 95% confidence interval for the population mean of the heights of the female students. [2 marks] The confidence interval (175.377 cm 177.223 cm) was calculated for the population mean of the heights of the male students. b) Determine the confidence level that was used to calculate this interval. Give your answer correct to 0.1%. [3 marks]
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 5 A confidence interval for a parameter is a range of values within which the (A) sample estimate of the parameter always lies. (B) sample estimate of the parameter never lies. (C) parameter always lies. (D) parameter never lies.
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 13 (4 marks) The wait time for customers put on hold when calling complaint departments is assumed to be normally distributed. A company claims that the mean wait time for their customers is 7.6 minutes. The following data represents the wait time (minutes) from a random sample of 12 customers who called the complaint department of this company. a) Determine the mean of this data. [1 mark] The standard deviation of this data is calculated to be 2.384 minutes. b) Use an approximate 95% confidence interval for the mean to evaluate the reasonableness of the company’s claim. Justify your decision using mathematical reasoning. [3 marks]
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
QUESTION 19 (7 marks) The height of Year 9 students at a school is assumed to be normally distributed with a population mean height of uw cm. A teacher at the school measured the height of all the students in her Year 9 class. This data was used to calculate an approximate 95% confidence interval for u of (163.7 166.9) cm. The teacher repeated the procedure using data from another Year 9 class. Although this class had the same number of students its data produced an approximate 95% confidence interval for uw of (167.8 172.4) cm. Using the same data the teacher recalculated the approximate confidence intervals for for each class using a confidence level of x%. She observed that the upper bound of the confidence interval from her Year 9 class now equalled the lower bound of the confidence interval from the other Year 9 class. Determine the value of x. Give your answer rounded to one decimal place.
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