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Chapter3b

Chapter3b - Chapter 3 A Closer Look at Assumptions Stat...

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Chapter 3: A Closer Look at Assumptions Stat 3022 Section 001 Fall 2011 September 19, 2011 Stat 3022 (Section 001) Chapter 3 September 19, 2011 1 / 20

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Introduction Chapter 2 discussed the mechanics of using one-sample and two-sample t -procedures to perform statistical inference. Namely, t -tests and confidence intervals. We base these procedures on certain assumptions we have random sample(s), representative of population(s) data come from Normal population(s) in two-sample settings, samples are independent in two-sample settings, we have equal variance ( σ 1 = σ 2 = σ ) In practice, these assumptions are usually not strictly met. When are these procedures still “appropriate”? Stat 3022 (Section 001) Chapter 3 September 19, 2011 2 / 20
Making it Rain – A Randomized Experiment Data collected in southern Florida between 1968–1972 to test hypothesis that massive injection of silver iodide (AgI) into cumulus clouds can lead to increased rainfall. This process is called “cloud seeding”. Over 52 days, either seeded a target cloud or left it unseeded (as control). Randomly assigned treatment. Researchers were blind to the treatment – pilots flew through cloud every day, whether treatment or control, and mechanism in plane either seeded the cloud or left it unseeded. Question : Did cloud seeding have an effect on rainfall? If so, how much? Stat 3022 (Section 001) Chapter 3 September 19, 2011 3 / 20

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Graphical Summaries > boxplot(RAINFALL~TREATMENT,ylab="Rainfall (acre-feet)") SEEDED UNSEEDED 0 500 1000 1500 2000 2500 Rainfall (acre-feet) Stat 3022 (Section 001) Chapter 3 September 19, 2011 4 / 20
Graphical Summaries > par(mfrow=c(2,1)) > hist(RAINFALL[TREATMENT=="SEEDED"], main="Seeded", xlab="Rainfall (acre-feet)", xlim=c(0,3000), breaks=10, col="gray") > hist(RAINFALL[TREATMENT=="UNSEEDED"], main="Unseeded", xlab="Rainfall (acre-feet)", xlim=c(0,3000),breaks=8, col="gray") Seeded Rainfall (acre-feet) Frequency 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 12 Unseeded Rainfall (acre-feet) Frequency 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 Stat 3022 (Section 001) Chapter 3 September 19, 2011 5 / 20

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Numerical Summaries and Interpretations > numSummary(RAINFALL, statistics=c("mean","sd"), groups=TREATMENT) mean sd n SEEDED 441.9846 650.7872 26 UNSEEDED 164.5885 278.4264 26 Graphical and numerical summaries indicate that rainfall tended to be greater on seeded days. However, there are problems with our necessary assumptions: both distributions are very skewed both distributions have outliers variability is much greater in the seeded group than in the unseeded group Can we use our usual t -tools to analyze these data?
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Chapter3b - Chapter 3 A Closer Look at Assumptions Stat...

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