Use normal distributions to approximate binomial distributions

Learning Objectives Identify the conditions under which a normal distribution can be used to approximate a binomial distribution. Calculate the mean (μ) and standard deviation (σ) for a binomial distribution. Apply the continuity correction to account for approximating a discrete distribution with a continuous one. Convert binomial outcomes into z-scores using the calculated mean and standard deviation. Use a standard normal distribution table (z-table) or calculator to find probabilities. Solve multi-step problems involving the normal approximation to the binomial distribution. How can a company predict that out of 10,000 products, between 480 and 520 will be defective, without testing every single one? 🏭 Let's find out! In this tutorial, you will learn a powerful st...

TermDefinitionExample Binomial DistributionA discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and a constant probability of success.The number of 'heads' obtained when a fair coin is flipped 20 times. Here, n=20, p=0.5. Normal DistributionA continuous probability distribution represented by a symmetric, bell-shaped curve. It is defined by its mean (μ) and standard deviation (σ).The distribution of adult human heights, which cluster around an average height and become less common at the extremes. Discrete vs. Continuous VariableA discrete variable can only take on specific, countable values (e.g., integers). A continuous variable can take on any va...

Conditions for Normal Approximation np \ge 5 \quad \text{and} \quad nq \ge 5 Before approximating, you must verify that the sample size (n) is large enough. Calculate the expected number of successes (np) and failures (nq), where q = 1-p. If both are at least 5, the binomial distribution is sufficiently symmetric to be approximated by a normal distribution. Binomial Mean and Standard Deviation \mu = np \quad \text{and} \quad \sigma = \sqrt{npq} These formulas convert the parameters of the binomial distribution (n, p) into the parameters of the approximating normal distribution (mean μ and standard deviation σ). Z-Score Formula Z = \frac{x - \mu}{\sigma} This formula standardizes a value 'x' from your normal distribution into a Z-score. This allows you to us...