April 18, 2009 - Class Notes

Class Lecture Notes for April 18, 2009 (talking points)

A. Introduction to Normal Distributions

A continuous probability distribution is the probability distribution of a continuous random variable.

A normal distribution is a continuous probability distribution describing the behavior of a normal
random variable. A normal probability distribution has a graph that is symmetric and bell shaped. Its mean, median, and mode are equal and determine the axis of symmetry. The graph of a normal distribution is defined for all numbers on the real number line. As the random variable x moves further and further from the mean—in either direction—the graph of the normal distributions approaches butnever touches the x-axis.

Between the points x = μ – σ and x = μ + σ, the graph is curved downward. To the left of x = μ – σ and to the right of x = μ + σ, the graph is curved upward. The points x = μ – σ and x
= μ + σ are called inflection points.


The normal curve, or the bell-curve, is the graph of a normal distribution.
Properties of a normal distribution can be listed as follows:
• The mean, median, and mode are equal.
• The normal curve is bell-shaped and symmetric about the mean, μ.
• The total area under the curve is equal to 1.
• The normal curve approaches the x-axis but never touches the axis as it extends further and
further away from the mean.
• At the center of the curve, between (μ − σ) and (μ + σ), the graph curves downward. The graph
curves upward to the left of (μ − σ) and to the right of (μ + σ).

Properties of a standard normal distribution can be listed as follows:
• The standard normal curve is bell shaped and symmetric about 0.
• The total area under the curve is equal to 1.
• The standard normal curve approaches the x-axis but never touches the axis as it extends
further and further away from the mean.
• At the center of the curve, between –1 and 1, the graph curves downward. The graph
curves upward to the left of –1 and to the right of 1.
• The cumulative area is close to 0 for z-scores close to z = −3.49.
• The cumulative area increases as the z-scores increase, but it never exceeds 1.
• The cumulative area for z = 0 is 0.5000.
• The cumulative area is close to 1 for z-scores close to z = 3.49.

B. Normal Distributions: Finding Probabilities
The probability of a normally distributed random variable x can be calculated using the following
guidelines:

Step 1: Find the x-values of the upper and lower bounds of the given interval.
Step 2: Convert the x-values to z-scores using the formula:
Step 3: Sketch the standard normal curve and shade the appropriate area under the curve.
Step 4: Find the area by following the same directions as given in the table for the standard normal probability distribution.


C. Normal Distributions: Finding Values of the Random Variable x, Given the Standard Normal
Random Variable z
Find variable x-values within areas under the standard normal curve:
Step 1: Determine the position of the area corresponding to the given probability.
Step 2: Find the corresponding z-scores for the area using the standard normal distribution table.
Here, you may have two cases:

• Area to the left of z
• Area to the right of z
Step 3: Transform the z-score to an x-value, using the formula: x = μ + zσ
D. Sampling Distributions and the Central Limit Theorem
A sampling distribution is the probability distribution of a sample statistic that is formed when
samples of size n are repeatedly taken from a population.
If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample
means.
The properties of sampling distributions of sample means can be listed as follows:
1. The mean of the sample means μ x is equal to the population mean.
μ x = μ
2. The standard deviation of the sample means σ x is calculated by dividing the population
standard deviation σ by the square root of n—the sample size.
n
x
σ
σ =
σ x is also known as the standard error of the mean.


The Central Limit Theorem is an important concept in inferential statistics. It enables you to make inferences about a population mean based upon sample statistics.