91Ó°ÊÓ

The binomial distribution is a specific type of probability distribution that describes the outcomes of a fixed number of independent and identical trials, each having two possible outcomes: success or failure. In the context of our exercise, each component could be either defective (success, in this case) or non-defective (failure). The key parameters critical to a binomial distribution are \( n \) and \( p \), where \( n \) is the number of trials, and \( p \) is the probability of success in a single trial. For our problem:

• \( n = 400 \) (the total number of components)
• \( p = 0.01 \) (the probability that a component is defective)

The random variable \( X \), representing the number of defective items, follows a binomial distribution with these parameters. Understanding this model is crucial as it helps in calculating probabilities related to our defective components sampling.

Normal approximation is a valuable technique for simplifying calculations in binomial distributions, especially when the sample size \( n \) is large, and the probability \( p \) is small. This makes the binomial distribution resemble a normal distribution.
For our problem:

• The mean of the binomial distribution is calculated using \( \mu = np = 400 \times 0.01 = 4 \).
• The variance is \( \sigma^2 = np(1-p) = 400 \times 0.01 \times 0.99 = 3.96 \).
• The standard deviation is determined as \( \sigma = \sqrt{3.96} \approx 1.9899 \).

These steps demonstrate how the binomial distribution transitions into a roughly bell-shaped curve (the normal curve) under large sample sizes, allowing for easier probability calculations.

When transforming a binomial distribution into a continuous normal distribution, a continuity correction is necessary. A binomial variable is discrete, meaning it takes integer values, but when approximated with a normal distribution, we deal with continuous values.
Continuity correction involves adjusting for this shift:

• For calculating \( P(X \leq 2) \), we use \( P(X \leq 2.5) \) to more accurately reflect the discrete nature by covering the half-unit interval.
• Similarly, for \( P(X \geq 7) \), we apply the correction to consider \( P(X \geq 6.5) \).

This slight modification helps in aligning the continuous normal approximation more closely with the original discrete binomial distribution, enhancing accuracy in probability estimation.

The standard normal distribution, often symbolized by \( Z \), is a key concept used to determine probabilities once a distribution has been approximated as normal. It is a normal distribution with a mean of 0 and a standard deviation of 1.
To convert a binomial variable to a standard normal variable, we calculate a z-score using the formula:

• \( z = \frac{x - \mu}{\sigma} \)

For example:

• To find \( P(X \leq 2.5) \), calculate \( z = \frac{2.5 - 4}{1.9899} \approx -0.753 \).
• To find \( P(X \geq 6.5) \), calculate \( z = \frac{6.5 - 4}{1.9899} \approx 1.258 \).

Using the standard normal distribution table, these z-scores help us find the probabilities associated with our desired events, offering a streamlined method for solving such statistical problems.