Please do question 1 to 3

Question 1 [Total = 10 marks] Correct your final answers to 4 decimal places whenever appropriate. The lifetime of light bulbs produced by a manufacturer are assumed to be normally distributed with a mean of 280 days and a standard deviation of 50 days. (a) Find the probability that a randomly selected light bulb has a lifetime shorter than 302 days. (2 marks) (b) Suppose a random sample of 100 light bulbs is drawn, find the probability that at least 59 of the selected light bulbs have lifetime shorter than 302 days by using continuity correction method (4 marks) (c) Suppose a random sample of 100 light bulbs is drawn. Find the probability the mean lifetime of the sample lies between 282 days and 290 days. (4 marks) Question 2 [Total = 10 marks] The following data represent the lengths (in cm) of a sample of 10 metal strips produced by a machine. It is given that the lengths of the metal strips produced by the machine are normally distributed. 48.3 48.6 49.2 49.8 50.0 50.2 50.4 50.4 50.5 50.7 (a) Construct a 95% confidence interval for the mean length of the metal strips produced by this machine. Correct your final answers to 3 decimal places. (6 marks) (b) Find the minimum sample size required to estimate the true mean within 0.2 cm with 90% confidence given that the population standard deviation is 0.7 cm. (4 marks) Question 3 [Total = 10 marks] The government of a city implemented a new housing policy in 2018. Before the implementation, a study reported that 70% of the citizens supported the new policy. A researcher now selects a random sample of 200 citizens and these citizens are asked if they support the policy. (a) Would the sampling distribution of the sample proportion be approximated by a normal distribution? Show your calculations. (2 marks) (b) The government claims that the proportion of the citizens who support the policy has increased after the implementation. If 148 citizens of the 200 selected citizens support the policy, use the critical value approach to perform a hypothesis testing for the government's claim at a 0.01 level of significance. What conclusion can you draw from your result? Correct the test statistic to 2 decimal places. (8 marks)