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Question 2 (16 marks) The daily number of mobile phones sold is independent and has a normal distribution with mean 5000 units and standard deviation of 1600 units in a certain city. A marketing manager wants to examine the change in the daily number of mobile phones sold in the time following an online advertising on social media which was launched on 315 January 2021. On 16 randomly selected days in February and March in 2021, he finds that the mean daily number is 5500 and that the daily number is below 5000 units on 11 days. Assume that any variations in the market will change the mean of the distribution of the daily number of mobile phones sold only. (a) Find the 95% confidence interval for the mean daily number of mobile phones sold in February and March in 2021. (3 marks) (b) Using (a), determine whether online advertising on social media takes effect on the daily number of mobile phones sold. (1 mark) (c) If the margin of error in (a) is limited to be at most 400, find the least additional number of days that the marketing manager has to consider. (4 marks) (d) From April to November in 2021, the manager finds that the mean daily number of mobile phones sold on 48 randomly selected days is 7000 and that the daily number is below 5000 on merely 13 days. Using the data on all 64 days, (i) find the 95% confidence interval for the mean daily number of mobile phones sold, (ii) find the probability that the daily number of mobile phones sold is at least 5000 units, (iii) find the conditional probability that the daily number of mobile phones sold is at least 5000 units given that a day in either February or March is selected, and (iv) Using (d)(ii) and (d)(ii), explain whether the period of time and the daily number of mobile phones sold are independent. (8 marks) Remarks to students: Unless otherwise specified, parts are independent but dependent on the question statement. For example, (a), (b), (c) and (d) are independent. Sub-parts are independent of each other but dependent on the question statement and the part under which they are. For example, (d)(i), (d)(ii) and (d)(iii) are independent but dependent on the question statement and (d). However, in (b) and (d)(iv), it is specified that the results in (a), and (d)(ii) and d(iii) are referred to, respectively. Hence, the (b) is dependent on (a) and (d)(iv) is dependent on (d)(ii) and (d)(iii)