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UW-Madison MATH 141 - 6D -Statistical Inference

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6D: Statistical Inference6D: Statistical InferenceA set of measurements or observations in a statistical study is said to be statistically significant if it is unlikely to have occurred by chance.Example 1: The Swedish flu lasts about a week; a new drug is tested on 20 subjects (10 in the control group, 10 in the treatment group).Control group: mean duration of flu = 7.0 days;standard deviation = 2.0 days.Treatment group:mean duration of flu = 6.5 days;standard deviation = 2.0 days.Is this result statistically significant?Probably not; random fluctuations on the order of 2.0 days are bigger than experimental differences of 0.5 days.Example 2: The German flu lasts almost exactly one week; a new drug is tested on 20subjects (10 in the control group, 10 in the treatment group).Control group: mean duration of flu = 7.0 days;standard deviation = 0.2 days.Treatment group:mean duration of flu = 6.5 days;standard deviation = 0.2 days.Is this result statistically significant?Probably; random fluctuations on the order of 0.2 days are smaller than experimental differences of 0.5 days.However, such a small difference in the duration of an illness (6.5 vs. 7.0 days) might not be deemed by doctors to be clinically significant even if it is statistically significant. That is, statistics may convince us that the effect is real, but if the effect is too small, who cares?Quantifying statistical significance:* If the probability of an observed difference occurring by chance is 1 in 20or less, we say the difference is statistically significant at the 0.05 level.* If the probability of an observed difference occurring by chance is 1 in 100 or less, we say the difference is statistically significant at the 0.01 level.Margin of Error and Confidence Interval (the rough-and-ready version):Suppose you draw a sample of size n from a large population and measure that a proportion of p of the sample has a certain trait. Then you can be 95% confident that the proportion of the population with this trait is p plus-or-minus 1/sqrt(n).This rule works best when p stays far away from 0.0 (i.e., 0%) and 1.0 (i.e., 100%).Example: Out of a population of 10,000 people, you sample 100 and find that 70 (70% of the sample) have brown eyes. Then you can be 95% confident that the proportion of brown-eyed people in the population is between 0.70 – 1/sqrt(100) = 0.70 – 0.10 = 0.60and0.70 + 1/sqrt(100) = 0.70 + 0.10 = 0.80;i.e., you can be 95% confident that somewhere between 60% and 80% of the population is brown-eyed (margin of error = 10%).If we wanted to reduce the margin of error by a factor of ten (i.e., reduce the margin of error from 10% to 1%), you would have to increase the sample size by a factor of ten squared, or 100.Null and Altnernative Hypotheses:A null hypothesis :claims a specific value for a population parameter (“70% of Americans have brown eyes”); orclaims that there is no difference between the population parameters of two different populations(“The average intelligence of blue-eyed people equalsthe average intelligence of brown-eyed people”), orclaims that two variables are not correlated (“People who smoked between the ages of 20 and 30 are no more (and no less) likely to have lung cancer at age 60 than people who did not smoke between the ages of 20 and 30”);or ...We’ll focus on null hypotheses of the first sort.Null hypothesis: population parameter = claimed value.The alternative hypothesis is the claim that is accepted if the null hypothesis is rejected.Two possible outcomes of a hypothesis test:* We reject the null hypothesis, and say“We have evidence that supports thealternative hypothesis”; or,* We do not reject the null hypothesis, andsay “We lack sufficient evidence to support the alternative hypothesis.”Courtroom analogy: Murder trials, the burden of proof, and the standard of reasonable doubt.Null hypothesis: “X did not murder Y.”Alternative hypothesis: “X murdered Y.”Outside of TV dramas, it does not usually happen that the defense attorney can prove that the defendant is innocent.The prosecutor argues that the likelihood of innocence is incredibly low.The defense attorney argues that the likelihood of innocence is not that low. In unit 6D, we are not measuring innocence or guilt, but rather a mean (e.g., the average mileage of a new line of cars) or aproportion (e.g., the proportion of those carswith defective brakes).The outcome of a hypothesis test depends onhow stringent we want to be.We decide the outcome of a hypothesis test by comparing the actual sample result (meanor proportion) to the range of results we would expect if the null hypothesis were true.Let p be the chance that we would get the observed result (or a result even more at odds with the null hypothesis) if the null hypothesis were true. * If p is less than 0.01, the test is significantat the 0.01 level. The test offers strongevidence for rejecting the null hypothesis(and accepting the alternativehypothesis).* If p is greater than 0.01 but less than 0.05, the test is significant at the 0.05 level. The test offers moderate evidencefor rejecting the null hypothesis.* If p is greater than 0.05, the test is notsignificant. The test does not providesufficient grounds for rejecting the


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