Continuous Random Variables Continuous Random Variables Chapter 6

Overview 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 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.

Remember  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 6 OUR FOCUS

Outline 6.1 Introduction to the Normal Curve 6.2 Reading a Normal Curve Table 6.3 Finding the Probability using the Normal Curve 6.4 Find z-values using the Normal Curve 6.5 Find t-Values using the Student t- distribution

Section 6.1 Introduction to the Normal Curve Objectives: ◦Identify the properties of a normal distribution

What is a Normal Distribution? A normal distribution is a continuous, symmetric, bell-shaped distribution of a variable

Any particular normal distribution is determined by two parameters ◦Mean,  ◦Standard Deviation, 

Properties of the Theoretical Normal Distribution A normal distribution is bell-shaped and is symmetric Symmetry 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 Normal 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 or 99.7%. (Empirical Rule)

Introduction

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 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 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 Normal 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)

Examples Find area under the standard normal distribution curve ◦Between 0 and 1.66 ◦Between 0 and ◦To the right of z = 1.10 ◦To the left of z = ◦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

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