Questions: Suppose that 93% of adults with allergies report symptomatic relief with a specific medication. Suppose the medication is given to 21 randomly selected new patients with allergies. Round to the fourth for all answers. a) If we randomly selected 21 new patients over and over again, on average how many would we expect to have symptomatic relief? b) Find the variance of the distribution, c) Find the standard deviation of the distribution.

Solution Steps

The problem describes a binomial distribution where:

• The probability of success (symptomatic relief) is \( p = 0.93 \).
• The number of trials (patients) is \( n = 21 \).

The binomial distribution is characterized by:

• Mean: \( \mu = n \cdot p \)
• Variance: \( \sigma^2 = n \cdot p \cdot (1 - p) \)
• Standard deviation: \( \sigma = \sqrt{n \cdot p \cdot (1 - p)} \)

The expected number of patients with symptomatic relief is the mean of the binomial distribution: \[ \mu = n \cdot p = 21 \cdot 0.93 = 19.53 \]

The variance of the binomial distribution is: \[ \sigma^2 = n \cdot p \cdot (1 - p) = 21 \cdot 0.93 \cdot (1 - 0.93) = 21 \cdot 0.93 \cdot 0.07 = 1.3659 \]

The standard deviation is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{1.3659} = 1.1687 \]

Final Answer