Homework ReviewRandom VariablesThe Uniform DistributionA Non-uniform DistributionAssignmentAnswers to Even-numbered ExercisesContinuous Random VariablesLecture 22Section 7.5.4Robb T. KoetherHampden-Sydney CollegeMon, Feb 22, 2010Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 1 / 36Outline1Homework Review2Random VariablesThe Uniform DistributionA Non-uniform Distribution3Assignment4Answers to Even-numbered ExercisesRobb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 2 / 36Outline1Homework Review2Random VariablesThe Uniform DistributionA Non-uniform Distribution3Assignment4Answers to Even-numbered ExercisesRobb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 3 / 36Homework ReviewExercise 6.25, page 380.Machine A makes parts whose lengths are approximately normallydistributed with a mean of 4.6 mm and a standard deviation of 0.1 mm.Machine B makes parts whose lengths are approximately normallydistributed 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 fromMachine 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 MachineB, 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 / 36Homework ReviewExercise 6.25, page 380.(a) Draw the distributions for the lengths of parts under H0and underH1. For both sketches, label the x-axis from 4.2 to 5.2 by 0.1. Besure to include all important features.4.44.64.85.05.21234Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 5 / 36Homework ReviewExercise 6.25, page 380.(a) Draw the distributions for the lengths of parts under H0and underH1. For both sketches, label the x-axis from 4.2 to 5.2 by 0.1. Besure to include all important features.4.44.64.85.05.21234Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 5 / 36Homework ReviewExercise 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 tothis p-value and clearly label the region as such.4.44.64.85.05.21234Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 6 / 36Homework ReviewExercise 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 tothis p-value and clearly label the region as such.4.44.64.85.05.21234Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 6 / 36Homework ReviewExercise 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 / 36Homework ReviewExercise 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 / 36Homework ReviewExercise 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 / 36Homework ReviewExercise 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 / 36Outline1Homework Review2Random VariablesThe Uniform DistributionA Non-uniform Distribution3Assignment4Answers to Even-numbered ExercisesRobb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 8 / 36Random VariablesDefinition (Random variable)A random variable is a variable whose value is determined by theoutcome of a random process.Definition (Discrete random variable)A discrete random variable is a random variable whose set of possiblevalues is a discrete set.Definition (Continuous random variable)A continuous random variable is a random variable whose set ofpossible values is a continuous set.Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 9 / 36Continuous Probability Distribution FunctionsDefinition (Continuous Probability Distribution Function)A continuous probability distribution function, or pdf, for a randomvariable X is a continuous function with the property that the areabelow the graph of the function between any two points a and b equalsthe probability that a ≤ X ≤ b.Remember,AREA = PROPORTION = PROBABILITYRobb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 10 / 36Outline1Homework Review2Random VariablesThe Uniform DistributionA Non-uniform Distribution3Assignment4Answers to Even-numbered ExercisesRobb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 11 / 36ExampleThe TI-83 will return a random number between 0 and 1 if weenter 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 therand function.Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 12 / 36ExampleWhat is the probability that the random number is at least 0.3?f(x)x0 11Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 13 / 36ExampleWhat is the probability that the random number is at least 0.3?f(x)x0 110.3Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 14 / 36ExampleWhat is the probability that the random number is at least 0.3?f(x)x0 110.3Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 15 / 36ExampleWhat is the probability that the random number is at least 0.3?f(x)x0 110.3Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010 16 / 36ExampleWhat is the probability that the random number is at least 0.3?f(x)x0 110.3Area = 0.7Robb T. Koether (Hampden-Sydney College) Continuous Random Variables Mon, Feb 22, 2010
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