Continuous Random Variables Lecture 22 Section 7 5 4 Robb T Koether Hampden Sydney College Mon Feb 22 2010 Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 1 36 Outline 1 Homework Review 2 Random Variables The Uniform Distribution A Non uniform Distribution 3 Assignment 4 Answers to Even numbered Exercises Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 2 36 Outline 1 Homework Review 2 Random Variables The Uniform Distribution A Non uniform Distribution 3 Assignment 4 Answers to Even numbered Exercises Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 3 36 Homework Review Exercise 6 25 page 380 Machine A makes parts whose lengths are approximately normally distributed with a mean of 4 6 mm and a standard deviation of 0 1 mm Machine B makes parts whose lengths are approximately normally distributed with a mean of 4 9 mm and a standard deviation of 0 1 mm Suppose that you have a box of parts which you believe are from Machine A but you re not sure You decide to test the hypotheses H0 The parts are from Machine A versus H1 The parts are from Machine B by randomly selecting one part from the box and measuring it Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 4 36 Homework Review Exercise 6 25 page 380 a Draw the distributions for the lengths of parts under H0 and under H1 For both sketches label the x axis from 4 2 to 5 2 by 0 1 Be sure to include all important features Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 5 36 Homework Review Exercise 6 25 page 380 a Draw the distributions for the lengths of parts under H0 and under H1 For both sketches label the x axis from 4 2 to 5 2 by 0 1 Be sure to include all important features 4 3 2 1 4 4 Robb T Koether Hampden Sydney College 4 6 4 8 Continuous Random Variables 5 0 5 2 Mon Feb 22 2010 5 36 Homework Review Exercise 6 25 page 380 b Suppose that you get a length of 4 8 mm i In your sketch for part a shade in the region that corresponds to this p value and clearly label the region as such Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 6 36 Homework Review Exercise 6 25 page 380 b Suppose that you get a length of 4 8 mm i In your sketch for part a shade in the region that corresponds to this p value and clearly label the region as such 4 3 2 1 4 4 Robb T Koether Hampden Sydney College 4 6 4 8 Continuous Random Variables 5 0 5 2 Mon Feb 22 2010 6 36 Homework Review Exercise 6 25 page 380 b ii Compute the p value for your test Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 7 36 Homework Review Exercise 6 25 page 380 b ii Compute the p value for your test The p value is 0 0228 Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 7 36 Homework Review Exercise 6 25 page 380 b ii Compute the p value for your test The p value is 0 0228 c What is your decision at the 0 01 level Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 7 36 Homework Review Exercise 6 25 page 380 b ii Compute the p value for your test The p value is 0 0228 c What is your decision at the 0 01 level The decision at the 0 01 level is to accept H0 Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 7 36 Outline 1 Homework Review 2 Random Variables The Uniform Distribution A Non uniform Distribution 3 Assignment 4 Answers to Even numbered Exercises Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 8 36 Random Variables Definition Random variable A random variable is a variable whose value is determined by the outcome of a random process Definition Discrete random variable A discrete random variable is a random variable whose set of possible values is a discrete set Definition Continuous random variable A continuous random variable is a random variable whose set of possible values is a continuous set Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 9 36 Continuous Probability Distribution Functions Definition Continuous Probability Distribution Function A continuous probability distribution function or pdf for a random variable X is a continuous function with the property that the area below the graph of the function between any two points a and b equals the probability that a X b Remember AREA PROPORTION PROBABILITY Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 10 36 Outline 1 Homework Review 2 Random Variables The Uniform Distribution A Non uniform Distribution 3 Assignment 4 Answers to Even numbered Exercises Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 11 36 Example The TI 83 will return a random number between 0 and 1 if we enter rand and press ENTER These numbers have a uniform distribution from 0 to 1 Let X be the random number whose value is determined by the rand function Robb T Koether Hampden Sydney College Continuous Random Variables Mon Feb 22 2010 12 36 Example What is the probability that the random number is at least 0 3 f x 1 x 0 Robb T Koether Hampden Sydney College 1 Continuous Random Variables Mon Feb 22 2010 13 36 Example What is the probability that the random number is at least 0 3 f x 1 x 0 Robb T Koether Hampden Sydney College 0 3 Continuous Random Variables 1 Mon Feb 22 2010 14 36 Example What is the probability that the random number is at least 0 3 f x 1 x 0 Robb T Koether Hampden Sydney College 0 3 Continuous Random Variables 1 Mon Feb 22 2010 15 36 Example What is the probability that the random number is at least 0 3 f x 1 x 0 Robb T Koether Hampden Sydney College 0 3 Continuous Random Variables 1 Mon Feb 22 2010 16 36 Example What is the probability that the random number is at least 0 3 f x 1 Area 0 7 x 0 Robb T Koether Hampden Sydney College 0 3 Continuous Random Variables 1 Mon Feb 22 2010 17 36 Example What is the probability that the random number is between 0 25 and 0 75 f x 1 x 0 Robb T Koether Hampden Sydney College 0 25 0 75 Continuous Random Variables 1 Mon Feb 22 2010 18 36 Example What is the probability that the random number is between 0 25 and 0 75 f x 1 x 0 Robb T Koether Hampden Sydney …
View Full Document
Unlocking...