Question

At a recent meeting at the American College of Asthma, Allergy, and Immunology, researchers reported that allergy vaccinations are effective in treating sinusitis (inflammation of the membrane lining the facial sinuses) in individuals predisposed to allergies. Specifically, the researchers reported that 98% of individuals predisposed to allergies find relief from their sinusitis after allergy vaccinations. Suppose that this number is accurate and that 140 individuals predisposed to allergies are chosen at random and given allergy vaccinations. a. Estimate the number of individuals in the random sample who do not find relief from their sinusitis by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response. b. Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places. (If necessary, consult a list of formulas.)

Answer 1

a)
`E(X) = 2.8`

b)
`\sigma = 1.657`

Explanation:

For each individual, there are only two possible outcomes. Either they find relief, or they do not. This means that we can solve this problems using concepts of the binomial probability distribution.

Binomial probability:

n trials, with a probability of success p.

Mean:
`E(X) = np`

Standard deviation:
`\sigma = √(np(1-p))`

In this problem

98% percent of the individuals find relief after vaccinations.

2% do not find relief after vaccinations.

140 individuals are chosen, so
`n = 140`.

a. Estimate the number of individuals in the random sample who do not find relief from their sinusitis by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable).

In this case, a success is not finding relief, so
`p = 0.02`

`E(X) = np = 140*0.02 = 2.8`.

b. Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.

`\sigma = √(np(1-p)) = √(140*0.02*0.98) = 1.657`