Statistics 106: Ch. 7, Properties of the Normal Distribution

Finding Area to the right of an X value

Use the complement rule: Area = 1- (area to the left of x-value).

Notes on graphs/normal curves

Mode= high point go the graph of any distribution (because it is the number that appears the most frequently). Median=point that splits top 50% from bottom 50%. Mean= balancing point of the graph of the distribution. For a symmetric single peak curve mean=median=mode, so mean is also the high point and center.

Note on Probability Density Functions

The area under the graph of a PDF over an interval represents the probability of observing a value of the random variable in that interval.

Probability Density Function (PDF)

The equation or model used to determine the probability of a continuous random variable.

Notes about changing mean and standard deviation

Changing the mean moves the graph left and right along the x axis e.g. if the mean is changed from 0 to 3 the graph slides three spots to the right, but it does not change its shape. Changing the standard deviation changes how steep the graph is but it maintains its center.

Inflection Points

x=mean-1 standard deviation and x=mean+1 standard deviation are the inflection points on the normal curve, the points on the curve where the curvature of the of the graph changes. To the left and right of the inflection points the graph slopes up towards the middle, which is dome shaped.

Normal Curve

A curve used to describe continuous random variables that are said to be normally distributed.

Finding the value of a Normal Random Variable

1. Draw a normal curve and shade the desired area. 2. Use table V to find the z score that corresponds to the shaded area/percentage 3. Obtain the normal value from the formula x= mean+Z•(standard deviation).

Properties of the Normal Density Curve

1. It is symmetric about the mean 2. Because mean=median=mode, the curve has a single peak and the highest point occurs at x=mean 3. It has inflection points at mean-1 standard deviation and mean + 1 standard deviation. 4. The area under the curve is 1. 5. The area under the curve to the right of the mean=area under to the curve to the left of the mean (both are 1/2). 6. As x increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis. As x decreases without bound, the graph approaches, but never reaches the horizontal axis. 7. The empirical rule can be applied: 68% of the area under the normal curve is within one standard deviation, 95% is within 2 standard deviations, and approximately 99.7% of the data is within 3 standard deviations.

Normally Distributed

A continuous random variable is normally distributed, or has a NORMAL PROBABILITY DISTRIBUTION, if its relative frequency has the shape of a normal curve.

Standard Normal Random Variable

A variable whose mean is 0 and whose standard deviation is 1. Usually the result of using a z-score (because these are z scores) to transform a normal random variable with mean does not equal 0 and a standard deviation that does not equal 1. Standard normal random variables can be used in conjunction with table V to look up areas under a curve given a z score to the left of the x value in the distribution.

Z-Score

Allows us to transform the random variable X with mean mew and standard deviation sigma into a random variable z with mean 0 and standard deviation 1.

Probability Density Function

An equation used to compute probabilities of continuous random variables must satisfy two properties: 1. The total area under the graph of the equation over all possible values must equal one. 2. The height of the graph must be ≥ 0 for all possible values of the random variable (so the graph never actually meets the axis).

Uniform Probability Distribution

When any two intervals of equal length are equally likely, the random variable X is said to follow a uniform probability distribution.

Model

An equation, table, or graph used to describe reality.

Note on Continuous Random Variables

Since an infinite # of outcomes are possible for a CRV, the probability of observing one particular value is 0 (this is why these are done over intervals).

Area under a Normal Curve

Suppose that a random variable X is normally distributed with mean mew and standard deviation sigma. The area under the normal curve for any interval of values of the random variable x represents either: 1. The proportion of the population with the characteristic described by the interval of values, or 2. The probability that a randomly selected individual from the population will have the characteristic described by the interval.

Standardizing a Normal Random Variable

Suppose that the random variable X is normally distributed with mean mew and standard deviation sigma, then the random variable is normally distributed with mean=0 and standard deviation=1. The random variable Z is said to have the standard normal distribution. Z=(x-mean)/(standard deviation).