(a) In order to decide the quarantine period for people who are at risk having Covid-Pi virus, the Taskforce must know the incubation period of the virus, i.e., how long it causes a person to show symptom after he has contacted the virus. Worldwide Statistics shows that the incubation periods follow a Normal Distribution with a mean of 12 days and a standard deviation of 5 days.

(i) Find the incubation period that only 3% of the Covid-Pi virus will last beyond before showing symptom in human bodies.

(ii) The Taskforce takes a sample of 15 airport arrival passengers who are tested positive for Covid-Pi virus. Evaluate the probability that the mean of their virus incubation periods will be more than 10 days. Explain why this analysis can be conducted on a small sample size.

Note: In Q(a)(i) & (ii), you shall use relevant Statistics formula and Standard Normal Distribution table to derive the answers.

(b)The first vaccine that could fight the Covid-Pi virus was developed in February 2021 by Cosmos Pharmaceutical Limited (CPL). It is named the Conqueror. A sample data was collected on its effective duration in weeks, i.e., how long the immunity lasts in a human body after the vaccination.

Six months later, another company Delta Life Science (DLS) develops its vaccine named Destroyer. A sample data is collected. DLS Management claims that its Destroyer has longer effective duration than Conqueror. Note: Only one dose will be administered per person for either vaccine. The data is shown in the GBA Data file (see worksheet 'Vaccine Effectiveness'). Use Microsoft Excel to summarise separately the two (2) sets of data. Identify the locations and spreads of the samples, each with a measure that takes all data into consideration. Comment on their differences. Present each data in an appropriate type of diagram. Comment on the data distributions.

(c)Use the Excel summary output in Q1(b) to identify the 95% Confidence Interval (CI)estimate of the mean effective duration for each type of vaccine, and interpret the confidence intervals. Hence infer any differences in the effective durations of the two vaccines.

State the requirement in using t-distributions to construct the confidence intervals.Justify whether this requirement is met in both cases.