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 120 individuals predisposed to allergies are chosen at random and given allergy vaccinations. Answer the following. (If necessary, consult a list of formulas.) (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.

Let's analyze the problem step by step: ### Given: - **Total individuals (\( n \))**: 120 - **Probability of finding relief (\( p \))**: 98% or 0.98 - **Probability of **not** finding relief (\( q = 1 - p \))**: 2% or 0.02 We are interested in the number of individuals who **do not** find relief from their sinusitis after the vaccination. This scenario follows a **binomial distribution** where the random variable \( X \) represents the number of individuals not finding relief. ### (a) **Mean (Expectation) of \( X \)** The mean of a binomial distribution is given by: \[ \text{Mean} (\mu) = n \times q \] Plugging in the values: \[ \mu = 120 \times 0.02 = 2.4 \] **Answer to (a):** The estimated number of individuals who do not find relief is **2.4**. ### (b) **Standard Deviation of \( X \)** The standard deviation of a binomial distribution is calculated using: \[ \text{Standard Deviation} (\sigma) = \sqrt{n \times q \times p} \] Substituting the given values: \[ \sigma = \sqrt{120 \times 0.02 \times 0.98} = \sqrt{2.352} \approx 1.534 \] **Answer to (b):** The standard deviation of the distribution is approximately **1.534**. ### Summary - **(a)** The expected number of individuals not finding relief is **2.4**. - **(b)** The standard deviation is **1.534**. **Final Answer:** Problem a Answer: The expected number is \( 2.4 \). Problem b Answer: The standard deviation is \( 1.534 \).