91Ó°ÊÓ

When dealing with a binomial distribution, the normal approximation can be a handy tool. This approximation is most useful when the number of trials, denoted by \( n \), is large, and the probability of success, denoted by \( p \), is neither too close to 0 nor too close to 1. In our exercise, we're working with 200 steel shafts, and 10% of these are nonconforming. Given this is a large number, a normal approximation simplifies calculations.

A binomial distribution can be approximated by a normal distribution when \( np > 5 \) and \( n(1-p) > 5 \). In this example, \( np = 20 \) and \( n(1-p) = 180 \), which easily meet these criteria. The mean (\( \mu \)) and variance (\( \sigma^2 \)) of the binomial distribution are \( np \) and \( np(1-p) \) respectively. Thus, the normal distribution's parameters in this scenario are calculated as a mean of 20 and standard deviation calculated as the square root of 18. Now, instead of complex binomial probability calculations, we can use normal distribution tables, which are usually easier to interpret.

Probability calculation using normal approximation involves converting binomial data to a standard normal distribution problem. This involves z-scores, which are standardized scores that relate binomial counts to the standard normal distribution.

To find the probability that a variable \( X \) falls within a certain range, you will often compute a z-score using the formula: \[ z = \frac{x - \mu}{\sigma} \] Where:\( x \) is the value of \( X \) you're interested in, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.

In practice, probabilities are determined by checking z-score values against a standard normal distribution table or using technology to find the area under the curve for those z-scores. For instance, to find the probability of \( X \leq 30 \), we determine the z-score and then find the corresponding probability in the z-table. This approach converts an otherwise intricate set of binomial calculations into more manageable normal distribution queries.

A crucial factor when using normal approximation for discrete distributions like the binomial is the continuity correction. This adjustment offers a more accurate approximation. When converting from a discrete to a continuous distribution, small adjustments (usually ±0.5) are added to the value of \( X \) before calculating a z-score.

This means, for the probability \( P(X \leq 30) \), instead, calculate \( P(X \leq 30.5) \). By adding 0.5, you're correcting for the fact that a continuous distribution assumes data between whole numbers. It's like smoothing out the steps between the intervals of a staircase, which better approximates the cumulative nature of a standard normal distribution.

• For "at most" scenarios like \( P(X \leq 30) \), use \( 30.5 \) instead.


• For "less than" cases, such as \( P(X < 30) \), use \( 29.5 \) to represent just beneath the threshold.


• When considering "between" scenarios like \( P(15 \leq X \leq 25) \), apply continuity correction on both bounds: \( 14.5 \) and \( 25.5 \).

This correction helps ensure the approximation closely reflects the true probability, leading to more reliable solutions.