To get full marks for the following questions you need to convert the question from words to a mathematical expression (i.e. use mathematical notation), defining your events where necessary, and using correct probability statements. A polling company performed research on how well Australian state premiers were performing in light of the COVID-19 pandemic. Respondents form each state of Australia were asked for their level of approval for their particular state premier, with possible option of "approve", "don't know", and "disapprove". In addition to this their gender was recorded.

Among men, 40 approved, 45 disapproved and 10 said don't know.

Among women, 62 approved, 30 disapproved and 5 said don't know.

For the following questions, use A1 to define the event that someone approves of their state premier, A2 and A3 to define the events disapprove and don't know respectively. Let the event M represent the event that the respondent was male. Use the above sample counts to determine the probabilities asked for in the following questions.

(a) [2 marks] What type of variable is the level of approval for one's state premier? Carefully explain your reasoning.

(b) [2 marks] Construct a contingency table summarising the data collected.

(c) [2 marks] If we were to choose a male at random, what is the probability that they will approve of the state Premier's performance?

(d) [2 marks] What is the probability that a randomly selected Australian is a woman who approves of the state Premier's performance?

(e) [2 marks] What is the probability that a randomly selected Australian approves of their Premier's performance?

(f) [2 marks] A member of the NSW state government was made aware of the research and emphatically reported to the NSW state premier that the majority of voters approve of their performance. Comment on whether you agree with this party members assessment of the data.