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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 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 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 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 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 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 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 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 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 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 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 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 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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