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09 Hypothesis Testing

# 09 Hypothesis Testing - Hypothesis Testing D&B 9.1 9.2 9.3...

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Hypothesis Testing Distribution of the Mean Confidence Intervals

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Motivation { We wish to use the mean to help characterize the nature of a process { Example: The average time of arrival (hours after midnight) at the intensive care unit of a local hospital { We are also interested in the quality of the estimate { Ultimately, we wish to compare that measure (i.e. the mean) to measures from other settings or to some hypothetical level (that is, we may wish to test a hypothesis )
0 5 10 15 20 0 5 10 15 20 25 Number of Patients Simulation Based on Process Analyzed by Cox and Snell 1981

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Simulation Script { admit = round(runif(254,0,24)) { hist(admit,xlab = "Hours After Midnight",cex=1.2) { mtext(side=2,"Number of Patients",cex=1.2,line=3)
First Question: { What is the average arrival time for the population as estimated from a sample of size n? = = N i i X N Population 1 1 μ = = n i i x n x Sample 1 1

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The Population The Sample 0 5 10 15 20 0 5 10 15 20 25 Hours After Midnight Number of Patients 0 5 10 15 20 25 0123456 Hours After Midnight
The Sample The Mean (n=25) sample1 = sample(admit,25) mean(sample1) 11.76 As compared to population mean(admit) 12.40157 0 5 10 15 20 25 0123456 Hours After Midnight Number of Patients

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Second Question: { What is the quality of the estimate? { How reliable is the sample mean at representing the true population mean? { Does the estimate vary much under repeated sampling?
To address this, let’s sample a bunch of times to see how the mean varies… sample.means = rep(0,100) for(i in seq(100)) { sample.means[i]= mean(sample(admit,25)) } hist(sample.means)

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8 1 01 21 41 6 0 5 10 15 20 25 sample.means
8 1 01 21 41 6 0 1 02 03 0 sample.means

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8 1 01 21 41 6 0 1 02 03 04 05 0 sample.means
8 1 01 21 41 6 02 0 0 0 0 sample.means

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8 1 01 21 41 6 0 2 04 06 0 0 sample.means
What influences the quality of the estimate? { Assume random sampling { The inherent population variation will influence the variation in the mean { The sample size will also influence the variation in the mean { We cannot always control population variation, but we can control sample size and how we take our observations

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in the mean… { But we would like to do this without having to go out and take 100 samples of size 25. { Can we estimate this with one
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09 Hypothesis Testing - Hypothesis Testing D&B 9.1 9.2 9.3...

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