Continuous Random Variables Continuous Random Variables Chapter 6.
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Transcript of Continuous Random Variables Continuous Random Variables Chapter 6.
Continuous Random Continuous Random VariablesVariables
Chapter 6
OverviewOverview
This chapter will deal with the construction of discrete probability distributions by combining methods of descriptive statistics from Chapters 2 and 3 and those of probability presented in Chapter 4.
A probability distribution, in general, will describe what will probably happen instead of what actually did happen
Combining Descriptive Methods and Probabilities
In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect.
Why do we need probability Why do we need probability distributions? distributions?
Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results◦ Saleswoman can compute probability that she will make
0, 1, 2, or 3 or more sales in a single day. Then, she would be able to compute the average number of sales she makes per week, and if she is working on commission, she will be able to approximate her weekly income over a period of time.
RememberRemember
Can be assigned values such as 0, 1, 2, 3
“Countable” Examples:
Number of children Number of credit cards Number of calls received
by switchboard Number of students
Can assume an infinite number of values between any two specific values
Obtained by measuring Often include fractions
and decimals Examples:
Temperature Height Weight Time
Discrete Variables (Data)—Chapter 5
Continuous Variables (Data)---Chapter 6OUR FOCUS
OutlineOutline
6.1 Introduction to the Normal Curve6.2 Reading a Normal Curve Table6.3 Finding the Probability using the
Normal Curve6.4 Find z-values using the Normal
Curve6.5 Find t-Values using the Student t-
distribution
Section 6.1 Introduction to the Section 6.1 Introduction to the Normal Curve Normal Curve
Objectives: ◦Identify the properties of a normal distribution
What is a Normal Distribution?What is a Normal Distribution?
A normal distribution is a continuous, symmetric, bell-shaped distribution of a variable
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Any particular normal distribution is determined by two parameters◦Mean, ◦Standard Deviation,
Properties of the Theoretical Properties of the Theoretical Normal DistributionNormal Distribution
A normal distribution is bell-shaped and is symmetricSymmetry of the curve means that if you cut
the curve in half, the left and right sides are mirror images (the line of symmetry is x =
Bell shaped means that the majority of the data is in the middle of the distribution and the amount tapers off evenly in both directions from the center
There is only one mode (unimodal) Mean = Median = Mode
Properties of the Theoretical Properties of the Theoretical Normal DistributionNormal Distribution
The total area under a normal distribution is equal to 1 or 100%. This fact may seem unusual, since the curve never touches the x-axis, but one can prove it mathematically by using calculus
The area under the part of the normal curve that lies within 1 standard deviation of the mean is approximately 0.68 or 68%, within 2 standard deviations, about 0.95 or 95%, and within 3 standard deviations, about 0.997 or 99.7%. (Empirical Rule)
IntroductionIntroduction
Uniform Distribution*Uniform Distribution*
A continuous random variable has a uniform distribution if its values are spread evenly over the range of possibilities. ◦The graph of a uniform distribution results in a
rectangular shape. ◦A uniform distribution makes it easier to see two
very important properties of a normal distribution The area under the graph of a probability distribution is
equal to 1. There is a correspondence between area and
probability (relative frequency)
Example: Roll a dieExample: Roll a die
Experiment: Roll a die◦Create a probability distribution in table form◦Sketch graph◦Using the graph, find the following
probabilities: ◦P(5)◦P(a number less than 4)◦P(a number between 2 and 6, inclusive)◦P(a number greater than 3)◦P(a number less than and including 6)
A researcher selects a random sample of 100 adult women, measures their heights, and constructs a histogram.
Because the total area under the normal distribution is 1, there is a correspondence between area and probability
Since each normal distribution is determined by its own mean and standard deviation, we would have to have a table of areas for each possibility!!!! To simplify this situation, we use a common standard that requires only one table.
Standard Normal DistributionStandard Normal Distribution
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
Finding Areas Under the Standard Finding Areas Under the Standard Normal Distribution CurveNormal Distribution Curve
Draw a picture ALWAYS!!!!!!! Shade the area desired. Follow given directions to find area (aka
probability) using the calculator
Area is always a positive number, even if the z-value is negative (this simply implies the z-value is below the mean)
ExamplesExamples
Find area under the standard normal distribution curve ◦Between 0 and 1.66◦Between 0 and -0.35◦To the right of z = 1.10◦To the left of z = -0.48◦Between z =1.23 and z =1.90◦Between z =-0.96 and z =-0.36◦To left of z =1.31◦To the left of z =-2.15 and to the right of z
=1.62
AssignmentAssignment
Worksheet