Chapter 5.1 -- Introduction to Normal Distributions and the Standard Normal Distributions

Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1.

The normal curve approaches, but never touches the ____-axis as it extends farther and farther away from the mean

X

The normal curve is _____-shaped and symmetric about the mean

bell

The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the _____

mean

The normal curve is bell-shaped and symmetric about the ______

mean

Properties of a Normal Distribution

1) A continuous probability distribution for a random variable, X 2) The most important continuous probability distribution in statistics 3) The graph of a normal distribution is called the normal curve

Means and Standard Deviations

1) A normal distribution can have any mean and any positive standard deviation 2) This mean gives the location of the line of symmetry 3) The standard deviation describes the spread of the data

Chapter 5.1 Objectives

1) How to interpret graphs of normal probability 2) How to find areas under the standard normal curve

Finding Areas Under the Standard Normal Curve

1) Sketch the standard normal curve and shade the appropriate area under the curve 2) Find the area by following the directions for each case shown. a) To find the area to the left of z, find the area that corresponds to z in the Standard Normal Table b) To find the area to the right of z, use the Standard Normal Table to fund the area that corresponds to z. Then subtract the area from 1.

Properties of Standard Normal Distribution

1) The cumulative area is close to 0 for z-scores close to z=-3.49 2) The cumulative area increases as the z-scores increase 3) The cumulative area for z=0 is 0.5000 4) The cumulative area is close to 1 for z-scores close to z=3.49

More Properties of a Normal Distribution

1) The mean, median, and mode are equal 2) The normal curve is bell-shaped and symmetric about the mean 3) The total area under the curve is equal to 1 4) The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean 5) Between the mean-1 SD and the mean+1 SD (in the center of the curve), the graph curves downward. The points at which the curve changes from curving upward to curving downward are called inflection points

Z-Score

Any x-value can be transformed into a z-score by using the formula

The mean, median, and mode are _______

Equal

Continuous Random Variable

Has an infinite number of possible values that can be represented by an interval on the number line

Continuous Probability Distribution

The probability distribution of a continuous random variable