Question 2.

[12 marks] According to the Australian Bureau of Statistics, 6.2% of Australians are depressed. An estimated 2.5% of Australians are mothers with at least one child under 2 years old. In total, 0.5% of Australians are mothers with at least one child under 2 years old and are depressed.

(a) [5 marks] Using this information, construct a contingency table of probabilities of depression and being a mother of at least one child under 2 years old.

(b) [2 marks] What is the probability that a randomly selected Australian is depressed or is a mother of at least one child under 2 years old?

(c) [3 marks] If a randomly selected Australian is a mother of a child under 2 years old, what is the probability that she is depressed?

(d) [2 marks] For Australians, is being depressed independent of being a mother of a child under 2 years old? Explain your answer.

Question 3.

[12 marks] A company that markets user assembled furniture sells a computer desk that is advertised with the claim "less than an hour to assemble". The company through their own testing have found that the assembly times follow are normal distribution with mean and standard deviation of 77.45 and 25.87 minutes respectively.

(a) [2 marks] Using this information, find what proportion of customers will complete the build in less than one hour.

(b) [2 marks] One way the company could solve this problem would be to change the advertising claim. What assembly time should the company quote in order that 60% of customers succeed in finishing the desk within that time?

(c) [2 marks] Wishing to maintain the "less than an hour to assemble" claim, the company hopes that revising the instruction and labelling the parts more clearly can improve the 1 hour success rate to 60%. If the standard deviation stays the same, show using the formula for transforming normal random variables that the new mean time will need to be 53.455 minutes.

(d) [3 marks] Assume that the company improved their plans for building the furniture and achieved a mean time of 53.455 minutes. A consumer advocacy group wishes to conduct their own testing and randomly selects 10 people to assemble the desk. It is recorded whether each person is able to complete the build within an hour or not. What type of distribution will the number of people who complete the build on time in this sample follow if the company's claim is true? Explain your reasoning.

(e) [3 marks] Suppose that of the 10 randomly selected people asked to assemble the desk only 4 managed to complete the assembly within 1 hour.

One member of the team conducting this test says that clearly the company's claim, that 60% of people can complete the assembly within 1 hour, is incorrect as 4 out of 10 is not 60%. Another team member is not so sure. What do you think? Does this test provide sufficient evidence to refute the company's claim?

(Hint: If the company's claim is correct, what is the probability that of the 10 people selected, four or less completed the assembly within the hour?)