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Normal approximation is a technique used when dealing with a binomial distribution that has a large sample size. It allows us to use the normal distribution, which is easier to work with mathematically. For normal approximation to be suitable, two conditions must be satisfied: \(np \geq 5\) and \(n(1-p) \geq 5\). Here, \(n\) is the sample size, and \(p\) is the probability of success (in this case, someone having a disability).

When these conditions are met, the binomial distribution can be approximated by a normal distribution with mean \(\mu = np\) and standard deviation \(\sigma = \sqrt{np(1-p)}\). This method simplifies the computation of probabilities for various numbers of successes in a binomial experiment.

Using the normal approximation simplifies the steps involved in calculating probabilities, especially when dealing with cumulative probability scenarios. This is especially helpful when the sample size is large, and the calculations are more complex without this approximation.

The binomial distribution is a discrete probability distribution. It describes the number of successes in a fixed number of independent trials, each with the same probability of success. In the context of our example, a 'trial' would be checking if a person sampled has a disability, which is a simple yes or no response.

Important features of a binomial distribution are:

• The number of trials \(n\) is fixed.
• Each trial is independent of the others.
• The probability of success \(p\) is the same in each trial.
• The random variable \(X\) counts the number of successes in \(n\) trials.

This distribution is defined by two parameters: \(n\) and \(p\). It forms the basis for the approximation methods, like the normal approximation, which make computations more manageable when the sample size is large.

Standard deviation (\(\sigma\)) is a measure of the amount of variation or dispersion in a set of values. In a binomial distribution, it is derived from the probability of success \(p\) and the number of trials \(n\).

The formula for the standard deviation of a binomial distribution is given by:
\[ \sigma = \sqrt{np(1-p)} \] This formula helps us understand how much variability we can expect in the number of successes from the mean, \(\mu = np\). It gives a sense of the range in which most of our observed data points lie.

Understanding standard deviation is crucial when using normal approximation because it allows us to convert the binomial problem into a standard normal (or \(Z\)-score) problem. The \(Z\)-score helps in looking up probabilities from standard normal distribution tables.

Continuity correction is an adjustment applied when a discrete distribution, like binomial, is approximated by a continuous distribution, like normal. The correction helps bridge the gap between the discrete and continuous probabilities.

Since binomial data is discrete and normal data is continuous, we shift our value by 0.5 to either side when calculating probabilities. This ensures that the area under the normal curve more accurately reflects the probability of the discrete data.

For example, when finding the probability of more than 200 people having a disability, the exact calculation of \(P(X > 200)\) involves computing \(P(X \geq 201)\), which is corrected by calculating \(P(X > 200.5)\).

This adjustment is integral to getting precise probability calculations when using normal approximation for discrete data, ensuring we consider the entire interval of interest correctly.