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The binomial distribution describes the probability of having exactly 'k' successes in 'n' independent trials with the same probability 'p' of a success in each trial. In our example, we are examining 100 smokers (n=100) with an 80% chance (p=0.80) that any one started smoking before the age of 18. It's essential for problems involving multiple trials where there are only two outcomes: success or failure. The parameters of the binomial distribution are simple: the number of trials, n, and the probability of success, p. Start by calculating the mean and standard deviation of the binomial distribution. The mean, \( \( \mu = np \) \), tells us the expected number of successes, while the standard deviation, \( \( \sigma = \sqrt{np(1-p)}\)\), gives us the spread of our distribution. These parameters allow us to approximate probabilities using the normal distribution.

The normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve. By the Central Limit Theorem, for large 'n', the binomial distribution can be approximated to a normal distribution if np and n(1-p) are both greater than 5. This makes calculations easier because instead of summing up binomial probabilities, we can use well-known properties of the normal distribution. The normal distribution is defined by its mean, \( \( \mu \) \), and standard deviation, \( \( \sigma \) \), which we computed from the binomial distribution. This transformation simplifies finding probabilities for specific values or intervals.

A z-score represents the number of standard deviations a data point is from the mean. In the context of approximating the binomial distribution with the normal distribution, z-scores are used to standardize our values so that we can use standard normal distribution tables or functions. The formula is \( \( z = \frac{x - \mu}{\sigma} \) \). For example, if we want to find the probability that fewer than 70 out of 100 smokers started smoking before 18, we first calculate the z-score for 69.5 (applying continuity correction), which is \( \frac{69.5 - 80}{4} = -2.625 \). This z-score allows us to use standard normal distribution tables to find the cumulative probability.

Continuity correction is a technique used when a discrete distribution (like the binomial distribution) is approximated by a continuous distribution (like the normal distribution). This correction accounts for the fact that the normal distribution represents continuous data. To apply it, adjust the discrete x-values by ±0.5 before converting to z-scores. For instance, when calculating the probability that exactly 80 of the smokers started smoking before 18, use 79.5 and 80.5 instead of 80. This small adjustment makes the approximation more accurate. Continuity correction is especially important in boundary conditions, ensuring that the computed probabilities are as close as possible to the true binomial probabilities.