In: Statistics and Probability
Recently, mumps outbreaks have become more common, with many occurring among individuals 18-24 years of age living on college campuses. Two doses of the measles-mumps-rubella (MMR) vaccine are recommended for protection from mumps. Herd immunity refers to the proportion of individuals that must be immune to effectively prevent the spread of disease through a population. In order to prevent the spread of mumps, at least 96% of people in a community must have received two doses of the MMR vaccine.
a) Suppose that 94% of undergraduate students in the United States report having received two doses of the MMR vaccine. What is the probability that in one upperclassman House at MIT, enough students are vaccinated to achieve herd immunity? There are approximately 400 students in any house.
b) Calculate the probability that herd immunity is achieved in all 12 Houses.
c) Discuss the validity of the assumptions required to make the calculation in part i.
a)
The given situation can be modelled to a binomial distribution as follows.
We define getting a student who reports having received two doses of the MMR vaccine as success.
Suppose, random variable X denotes the number of students found to report that they received two doses of the MMR vaccine.
It is given that in order to prevent the spread of mumps, at least 96% of people in a community must have received two doses of the MMR vaccine. So, we have to calculate which is very much tedious. So, we use normal approximation of the Binomial distribution.
i.e.
= P(Z>1.684304) = 0.04606147 [Using R-code '1-pnorm(1.684304)']
b)
Immunity achievement of each house is independent from that of other houses.
So, probability that herd immunity is achieved in all 12 houses = (probability that herd immunity is achieved in a houses)12
c)
In part (a) we approximated the Binomial distribution to normal distribution. However, certain assumptions are required to be fulfilled to do such. These are as follows.
So, both the assumptions required to make the calculations in part (a) are valid.