(a) A gum manufacturer claims that, on average, the flavour of an entire packet of gum would last for more than 39 minutes. A quality controller selects a random sample of 55 packets of gum. She finds the average time for which the gum flavour lasts is 40 minutes, with a standard deviation of 5.67 minutes.

(i) Formulate and conduct a hypothesis test to validate the manufacturer's claim. Is the claim justified?

(ii) After a new technique to improve the lasting period of gum flavour was applied, the quality controller re-selected 60 packets of gum and found that the average time for which the gum flavour lasts is 45 minutes with a standard deviation of 3.15 minutes. Is there sufficient evidence to conclude that the new technique significantly increased lasting time?

(iii) Use a 95% confidence interval to validate whether the new technique significantly increased lasting time. Justify your answer.

(b) A random sample of 1000 voters was taken, and they were asked about their voting intentions on a two-party preferred basis in the lead up to the recent Federal election. A total of 610 people indicated that they would vote for Labor. Calculate the 95% confidence interval for the proportion of Australian voters who intended to vote Labor. Interpret the confidence interval and comment on the chances of the Coalition returning to power based on this opinion poll.