Binomial experiments and the binomial probability formula with mean and standard deviation, the normal distribution, the empirical rule and z-scores.
Binomial experiments
Examples: the number of free throws made out of \(5\); the number of defective items in a sample of \(10\); the number of correct guesses on a multiple-choice quiz.
The binomial formula
The combination counts which trials are the \(k\) successes; the powers give the probability of each such sequence.
The normal distribution
Many quantities (heights, measurement errors, test scores) follow a symmetric, bell-shaped curve: the normal distribution.
Z-scores
Z-scores let you compare values from different distributions.
A standard normal table (or a calculator) gives the area to the left of any \(z\). For example, \(z = 1\) is at about the \(84\)th percentile: \(50\% + 34\%\).
Practice
Quick check
A binomial distribution has \(n = 50\) and \(p = 0.4\). What is its mean?
\(\mu = np = 50 \cdot 0.4 = 20\).
Quick check
A value has a z-score of \(-2\). What does that mean?
A z-score counts standard deviations from the mean.
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