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The Normal Distribution. And the standard normal distribution (also known as the “bell curve”). The Mathematics of the Normal Distribution. The normal distribution is completely described by only two parameters: the mean μ and the standard deviation σ. Review: Binomial Distribution.
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The Normal Distribution And the standard normal distribution (also known as the “bell curve”)
The Mathematics of the Normal Distribution The normal distribution is completely described by only two parameters: the mean μ and the standard deviation σ
Review: Binomial Distribution The Binomial Distribution is the outcome of a Bernoulli trial (only two possible values) repeated n times, with probability p of success. In the notation, x is the number of successes.
Binomial Example P = 0.25 (correct answer from a, b, c or d) N = 10 questions What is the probability of getting exactly 3 questions correct?
Binomial Dist. >>> Normal Dist. The Normal Distribution is a continuous distribution, and is a limiting case of a (discrete) binomial distribution. As the number of trials n -> ∞, the binomial distribution will approach the normal distribution. Here n = 5; experiment simulated 1,000 times. The binomial random variable X = number of success out of n = 5 trials, with probability p = 0.1 of success, repeated 1,000 times.
Simulation Continued (n increases) P = 0.1, n = 10 Simulated 1,000 times. P = 0.1, n = 25 Simulated 1,000 times.
Simulation Continued (n increases) P = 0.1, n = 25 Simulated 1,000 times. P = 0.1, n = 50 Simulated 1,000 times.
Other Simulations P = 0.5, n = 10 Simulated 1,000 times. P = 0.6, n = 100 Simulated 1,000 times.
Determinants of the place and shape of the normal distribution
Normal Distributions: Changing μ or σ Same mean μ Different standard deviation σ Different mean μ Same standard deviation σ
Normal Distributions: Changing both μ and σ Different mean μand Different standard deviation σ
The Sigma (standard deviation) Rule revisited 68% of the results will be between + or – 1 standard deviation (σ) away from the mean (μ). 95% of the results will be between + or – 2 standard deviation (σ) away from the mean (μ).
The Sigma (standard deviation) Rule revisited 99.7% of the results will be between + or – 3 standard deviation (σ) away from the mean (μ).
How to use a z-score Standard normal (Z-score) tables can tell you the probability of getting between the z-score and the mean 0. * Alternatively, the standard normal table could tell you the probability of getting a score anywhere below the z-score.
Sample Standard Normal Table To economize on space, the standard normal table has the first digit of the z-score down the first column, and the second digit of the z-score across the first row. Example: z = 0.92 0.3212 0.50 Pr(z ≤ 0.92) = 0.50 + 0.3212 = 0.8212
Standard Normal Curve: Example * Note that if we looked at P (-2 ≤ z ≤ ++2) = 0.4772 x 2 ≈ 0.95 or 95%, which is the 2-sigma rule.
Standard Normal Curve: Example • Note that the probability of getting a z-score less than z = -1.8 is P(z ≤ -1.8) = 1 – 0.50 – 0.4641 = 0.0359.
Applying the Standard Normal Distribution • If the underlying distribution is approximately normal…… • … Transform the normal variable into a standard normal variable using its z-score. • Solve the problem as above.
Online Help is Readily Available Google “z-score calculator” to convert normal variables into standard normal variables; Google “z-score table” or “standard normal table” to get the associated probability. Example (not necessarily recommended, and certainly not the only one available) Z-score calculator: http://www.danielsoper.com/statcalc3/calc.aspx?id=22 Standard normal z-score table: http://regentsprep.org/Regents/math/algtrig/ATS7/ZChart.htm which shows the area to the left of the z-score.