Reported case of COVID-19

1. 2019 saw the first reported case of COVID-19. Scientists from the World Health Organization are tasked
   with creating protocols for quarantining infected patients. In particular, they must outline the length of
   time that patients are to be kept in quarantine before they are considered safe to release and no longer
   contagious to the public.
   Assume that it is known that the time to recovery has a mean of 14.36 days and variance of 6.82 days.
   Be sure to define clearly any variables and models you use.
   (a) Assuming that the time to recovery follows a Normal distribution, find the following:
   i. An infected patient is selected at random. What is the minimum number of days that they
   should be quarantined such that there is a 99.9% chance that they will be recovered before
   release? [3 marks]
   ii. 100 infected patients are selected at random. What is the probability that their mean time to
   recovery is no more than 15 days? [3 marks]
   iii. Suppose 21 patients on a cruise ship simultaneously and suddenly become infected with the
   virus. With only 2 weeks remaining on their voyage, what is the probability that all patients
   will be recovered before they return to port, as to not infect anybody on shore when they return
   on the 15th day? Assume that the patients’ recovery are independent of one another. [4 marks]
   (b) Assume the true distribution of time to recovery is unknown. Given this, consider the following:
   i. Explain one reason why the Normal distribution might be a poor representation of the true
   distribution of time to recovery? Briefly explain in 1 – 4 sentences. [2 marks]
   ii. Is the probability you calculated in part (1(a)ii) still accurate? Briefly explain in 1 – 4 sentences.
   [2 marks]
   iii. A random sample of 144 infected patients reveals that their mean time to recovery is 15.09
   days. Is the mean recovery time of these 144 patients unusual? Justify your answer using
   probabilistic evidence. [2 marks]
2. Suppose a medical doctor suspects that 98% cases of COVID-19 would show fever symptoms. In order
   to study whether fever is one of the most common characteristics of COVID-19, this doctor randomly
   collected data on 1,099 patients with laboratory-confirmed COVID-19 and the result showed that 966
   patients have experienced the fever.
   (a) Use the doctors data to construct a 99% confidence interval for the true proportion of confirmed
   COVID-19 patients who had fever. [3 marks]
   1
   (b) Write out the assumptions and conditions necessary for the interval you constructed in part 2a in
   the context of the question. [3 marks]
   (c) If we say “we are 99% confident the true proportion of confirmed COVID-19 patients who had fever
   is between the range of the interval you calculated in part (2a)”. Provide an interpretation of that
   sentence in the context of the question. [2 marks]
   (d) When planning to estimate a population proportion, the doctor needs to determine the appropriate
   sample size. Suppose the doctor doesn’t have any prior information about the proportion and
   desires the estimate to be correct to within 0.05 with 99% confidence, how large a sample size is
   needed? [3 marks]
3. A certain airline guarantees customers that the airline rarely loses passengers’ baggage. The public
   relations department claims that on those occasions when luggage is lost, 92% is recovered and delivered
   to its owner within 24 hours. An independent consumer group surveyed travelers on this airline and
   found that 145 out of 165 people who lost luggage received their missing bags within 24 hours. You
   will conduct a hypothesis test to see whether the data collected by the independent consumer group is
   different from the airline’s claim.
   (a) Identify the population of interest in the context of the question. [1 mark]
   (b) State the null and alternative hypotheses in the context of the question in words. [1 mark]
   (c) Determine/compute any conditions that must be valid to carry out the test in the context of the
   question. [2 marks]
   (d) Compute the test statistic. [2 marks]
   (e) Find the exact value or provide a range of values for the P-value. Sketch your model, label and
   shade the corresponding region (a sketch by hand is fine). [2 marks]
   (f) State your conclusion in the context of the question with a 5% significance level. [1 mark]
   (g) Based on your conclusion above, what type of error are you at risk of making? Explain your answer
   in the context of this question in 1 – 4 sentences. [2 marks]

Sample Solution

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