According to the National Center for Health Statistics, 22.4% of adults are smokers. A random sample of 300 adults is obtained.
b. In a random sample of 300 adults, what is the probability that at least 50 are smokers?

Verified step by step guidance

1. Identify the type of distribution to use. Since we are dealing with the number of smokers in a sample and each adult can be considered a Bernoulli trial (smoker or not), the number of smokers follows a Binomial distribution with parameters \(n = 300\) and \(p = 0.224\).
2. Define the random variable \(X\) as the number of smokers in the sample. Then, \(X \sim \text{Binomial}(n=300, p=0.224)\).
3. The problem asks for the probability that at least 50 adults are smokers, which can be written as \(P(X \geq 50)\).
4. Since calculating binomial probabilities directly for large \(n\) can be cumbersome, consider using a normal approximation to the binomial distribution. The mean and standard deviation of \(X\) are given by \(\mu = np\) and \(\sigma = \sqrt{np(1-p)}\) respectively.
5. Apply the continuity correction by calculating \(P(X \geq 50) \approx P\left(Y \geq 49.5\right)\) where \(Y\) is a normal random variable with mean \(\mu\) and standard deviation \(\sigma\). Then convert to the standard normal variable \(Z = \frac{Y - \mu}{\sigma}\) and find the corresponding probability using standard normal distribution tables or software.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Here, each adult either is a smoker (success) or not, with probability 0.224. The distribution helps calculate probabilities for different counts of smokers in the sample.

Normal Approximation to the Binomial

When the sample size is large, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p). This simplifies probability calculations, especially for values like 'at least 50 smokers' in a sample of 300, where exact binomial calculations are complex.

Continuity Correction

When using the normal approximation for a discrete binomial variable, a continuity correction adjusts for the difference between discrete and continuous distributions. For example, to find P(X ≥ 50), we use P(X ≥ 49.5) in the normal distribution to improve accuracy.