2.  In a StatsNZ survey of 5000 New Zealand residents, survey respondents were asked which ethnicity they identified as and how many languages they spoke fluently. The following results were collected. MELAA stands for ‘Middle Eastern, Latin American or African’.

(a) Referring to the tree diagram on the following page give the appropriate probabilities that would be written on each branch where there are letters A to F. Give your answers as either fractions or decimals. Your answers should look like ‘A: xx/xx’ for a fraction or ‘A: 0.xxx’ for a decimal. Do not reproduce the tree diagram in your answer script. (Three probabilities are already written on the tree, so if we had asked you for P(Ethnicity is Asian) you would write ‘G: 503/5000’ or ‘G: 0.1006’)

(b) Using your probabilities from part 2a or using the table, give the following probabilities. You may wish to show your working for part 2(b)iii.

i. P(Ethnicity is Pasifika)

ii. P(Two or more languages spoken | Ethnicity is M¯aori)

iii. P(Two or more languages spoken ∩ Ethnicity is M¯aori)

iv. P(One language spoken)

v. P(Two or more languages spoken)

(c) Are the events ‘Ethnicity is M¯aori’ and ‘Two or more languages spoken’:

i. Mutually exclusive?

ii. Independent?

Justify your answer in each case (refer to week 3 notes if necessary). Use a mathematical equation to explain part 2(c)ii.

(d) Discuss briefly (in one or two sentences) whether you believe the variables Ethnicity and No. of languages are associated or not, justifying your answer.