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CH6-normal-distribution.md

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The normal distribution

objectives

identify distribution as symmetric or skewed
identify the properties of a normal distribution
find the area under standard normal distribution, by giving various z value
find the probabilities for a normal distributed variable by transforming it into a standard normal value
use the central limit theorem to solve problems
use the approximation to compute prob. for binomial variables

Normal distributions

The math function of normal distribution:

$$y = \frac{e^{-(X-\mu)^2/2\sigma^2}}{\sigma \sqrt{2\pi}}$$

  • Where

    • $e \sim 2.718$
    • $\pi \sim 3.14$
    • $\mu$ = population mean
    • $\sigma$ = population standard deviation

  • But in statistics, tables or technology is used for specific problem. normal distribution properties

1. bell-shaped
2. the mean, median, and mode are equal, and located at the center of the distribution
3.A normal distribution curve is unimodal (it has only one mode)
4. The curve is symmetric about the mean, which is equalvalent to saying that its shape is the same on both sides of a vertical line, there are no gaps or hole.
5. The curve is continuous; that there is no hole or gaps
6. The curve never touches the x axis but it gets closer
7. The total area under a normal distribution is equal to 1.00 or 100%
8. The area under the part of normal curve that lies within 1 standard deviation of the mean is appox. 0.68 or 68%, within 2 standard diviations, about 0.95, and within 3 standard diviation, about 0.997 or 997%.

The standard Normal Distribution

  • The standard normal distribution is a normal distribution with a mean of $0$ and standard diviation of $1$.
  • The standard normal distribution is the value under the curve indicate that proportion of area in each section.
  • Math func for the standard normal distribution

$$y = \frac{e^{-z^2/2}}{\sqrt{2\pi}}$$

  • All Normal distributed variables can be transformed into the standard normally distributed variable by using formula called standard score:

$$ z = \frac{X-\mu}{\sigma}$$

  • $X$: Value

  • $\mu$: Mean

  • $\sigma$: standard diviation

  • The $z$ value or $z-score$ is actually the number of standard diviations that a particular $X$ value is away from the mean.

How to find areas under the standard normal distribution curve?

  1. Draw the normal distribution curve and shade the area
  2. Find the appropriate figure in the Procedure Table and follow the directions given.

Procedure

  1. To the left of any z value: Look up the z value in the table and use the area give
  2. To the right of any z value: Look up the z value and subtract the area from 1.
  3. Between any two z value: Look up both z value and substract the corresponding areas.