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Hypothesis Tests for Means The context “Statistical significance” Hypothesis tests and confidence intervals The steps Hypothesis Test statistic Distribution Alpha, and the rejection region Result p-Values One-sided vs. two-sided tests Hypothesis tests for proportions

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The context PARAMETERS = population mean (unknown) = population SD (might be known) STATISTICS n = sample size x = sample mean s = sample SD (using n-1) ALSO 0 = conjectured value of

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Statistical significance We’re trying to decide whether is equal to 0. As usual we use x as an estimate of . Usually x is at least a little different from 0. But could the difference be due to random variation? IF YES – then we DO NOT REJECT the hypothesis that is really equal to 0. We say that x is not significantly different from 0. IF NO – then we REJECT the hypothesis that = 0. We say that x IS significantly different from 0.

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Hypothesis tests are just confidence intervals If we only cared about hypothesis tests for means, we could make this a lot simpler. Just construct a confidence interval for , based on n, x, s (or ) and your favorite confidence level C. If 0 is outside the confidence interval, then we reject the hypothesis that = 0. The significance level is = 1 – C. That’s all there is to it. So why all the complex ritual of a hypothesis test? Because there are other hypothesis tests, for other hypotheses (difference of two means, for example). For those tests, we need the ritual.

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Hypothesis Test for Cookbook using rejection regions 1. Choose hypotheses – H 0 and H A. 2. Define a test statistic. 3. Predict the distribution of the test statistic, assuming that H 0 is true. 4. Choose C and . Pick a rejection region. 5. Look at the observed value of the test statistic. Is it in the rejection region? If so, reject H 0.

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Hypothesis Test for Cookbook using rejection regions 1. Choose hypotheses – H 0 and H A. 2. Define a test statistic. 3. Predict the distribution of the test statistic, assuming that H 0 is true. 4. Choose C and . Pick a rejection region. 5. Look at the observed value of the test statistic. Is it in the rejection region? If so, reject H 0.

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Choose hypotheses Two-sided test: H 0 : = 0 H A : 0 One-sided tests: H 0 : = 0 H A : > 0 or H 0 : = 0 H A : < 0 Working rule: Always use two-sided tests.

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Hypothesis Test for Cookbook using rejection regions 1. Choose hypotheses – H 0 and H A. 2. Define a test statistic. 3. Predict the distribution of the test statistic, assuming that H 0 is true. 4. Choose C and . Pick a rejection region. 5. Look at the observed value of the test statistic. Is it in the rejection region? If so, reject H 0.

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Define a test statistic Choose or Do you know ? Maybe it comes with the null hypothesis. If so, use it.

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Hypothesis Test for Cookbook using rejection regions 1. Choose hypotheses – H 0 and H A. 2. Define a test statistic. 3. Predict the distribution of the test statistic, assuming that H 0 is true. 4. Choose C and . Pick a rejection region. 5. Look at the observed value of the test statistic. Is it in the rejection region? If so, reject H 0.

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Distribution of the test statistic ASSUME H 0 IS TRUE. Then (if you know ) z has a STANDARD NORMAL distribution. Or (if you’re using s) t has a “t” distribution with n-1 degrees of freedom.

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Hypothesis Test for Cookbook using rejection regions 1. Choose hypotheses – H 0 and H A. 2. Define a test statistic. 3. Predict the distribution of the test statistic, assuming that H 0 is true. 4. Choose C and . Pick a rejection region. 5. Look at the observed value of the test statistic. Is it in the rejection region? If so, reject H 0.

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(Standard normal case) The rejection region is a range (or double-range) of values of the test statistic that are (a) UNLIKELY if H 0 is true (b) roughly consistent with the alternative H A. The rejection region should have probability (given H 0 ). Two-sided case: z* /2 - z* /2 Rejection region consists of two parts, each with probability /2.

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Predicting the distribution If you’re using t, just use t-critical values. For the one-sided case: z* Rejection region probability , all in one tail.

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Chance of a Type I error Note: IF H 0 is actually true, then there is still a probability of that you will reject the null hypothesis. z* /2 - z* /2

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Chance of a Type I error There are two possible bad results: TYPE I ERROR (“act of commission”) – reject H 0, when H 0 is actually true. The probability of a Type I error is (given that H 0 is true) TYPE II ERROR (“act of omission”) – don’t reject H 0, when H 0 is actually false. The probability of a Type II error depends on the actual value of

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Hypothesis Test for Cookbook using rejection regions 1. Choose hypotheses – H 0 and H A. 2. Define a test statistic. 3. Predict the distribution of the test statistic, assuming that H 0 is true. 4. Choose C and . Pick a rejection region. 5. Look at the observed value of the test statistic. Is it in the rejection region? If so, reject H 0.

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Tradeoff High (say, 10%) then you have a good chance of having a statistically significant result, but it won’t impress anyone. MORE TYPE I ERRORS Low (say, 1%) then your significant results are more convincing, but you’ll have fewer of them. MORE TYPE II ERRORS Is there a way to avoid choosing in advance?

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Determine p-value The “p-value” is the answer to this question: What fraction of x ‘s are more extreme than the one you actually obtained? If H A : 0 this means, what fraction are further from zero than the value you obtained? If H A : > 0 this means, what fraction are more than the value you obtained? If H A : < 0 this means, what fraction are less than the value you obtained?

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Determine p-value Example: Do a test of H 0 : = 0 vs. H A : 0. Get test statistic z = 2.30. What’s the p-value? Probability of seeing 2.30 OR MORE: 0.0107 Probability of seeing 2.30 OR MORE EXTREME: 0.0214 p-value for 2-sided test: 0.0214 z=2.30 tail: 0.0107

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Determine p-value Keep it simple? p-value = (for 1-sided test with z) = 1 - NORMSDIST ( |z| ) (for 2-sided test with z) = 2 × (1-NORMSDIST(|z|)) (for 1-sided test with t) = TDIST ( |t|, n-1, 1 ) (for 2-sided test with t) = TDIST ( |t|, n-1, 2 ) df number of tails

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Determine p-value The p-value is the border between ’s for which we reject H 0 and ’s for which we do not reject H 0. REJECTION REGION VERSION: Pick , and the rejection region, in advance. In this story, the p-value is an afterthought. p-VALUE FIRST VERSION: Find the p-value first. Then if anyone has a favorite , you can… Reject H 0 if p < Do not reject if p > .

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Example: 1969 Draft Lottery Null hypothesis (informally): The numbers for the second half of the year were drawn randomly from the population 1, 2, …, 366. (Note: The mean of these numbers is 183.5, and their standard deviation is 105.6547. ) Null hypothesis (formally): H 0 : = 183.5 (and this is one of those cases where = 105.6547 comes with the null hypothesis) Alternative: H A : 183.5

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Example: 1969 Draft Lottery H 0 : = 183.5 H A : 183.5 0 = 183.5 = 105.6547 Experiment: n = 184, x = _________ Test statistic: p-value: Conclusion: REJECT H 0 (even at 1% significance level) 160.92 = - 2.898 0.00375

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Hypothesis tests for proportions PARAMETER p = population proportion STATISTICS n = sample size k = number of “hits” p = k / n = sample proportion

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Hypothesis tests for proportions Test statistic: (Minor subtlety: The distribution of the test statistic is based on H 0, so we use p 0 in the formula for SE. This is different from what we do in confidence intervals, but not by much.)

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Another example Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant?

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Another example Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant? Choose: H 0 : p = 0.50H A : p 0.50

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Another example Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant? Choose: H 0 : p = 0.50H A : p 0.50 Conditions? OK.

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Another example Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant? Choose: H 0 : p = 0.50H A : p 0.50 Conditions? OK. Distribution of p^, given H 0 : Normal, mean 0.50, SD=0.005

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Another example Our value of p^ is 0.51. That’s 2.0 SD’s above the mean. What fraction of p^ values would be further from zero than 0.51 ?

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Another example Our value of p^ is 0.51. That’s 2.0 SD’s above the mean. What fraction of p^ values would be further from zero than 0.51 ? ABOUT 4.5%, counting both tails. So, P-value is 0.045.

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Result of test Is a P-value of 0.045 good enough to reject H 0 ?

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Result of test Is a P-value of 0.045 good enough to reject H 0 ? If we choose = 0.05, then yes. But that’s a very mild test for such an extraordinary claim.

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Result of test Is a P-value of 0.045 good enough to reject H 0 ? If we choose = 0.05, then yes. But that’s a very mild test for such an extraordinary claim. If we pick = 0.05, then 5% of all our experiments will end in rejecting H 0, even though H 0 is true every time.

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Result of test Is a P-value of 0.045 good enough to reject H 0 ? If we choose = 0.05, then yes. But that’s a very mild test for such an extraordinary claim. If we pick = 0.05, then 5% of all our experiments will end in rejecting H 0, even though H 0 is true every time. So we should choose a lower value of . In this case, our result isn’t really “statistically significant.”

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Result of test Is a P-value of 0.045 good enough to reject H 0 ? If we choose = 0.05, then yes. But that’s a very mild test for such an extraordinary claim. If we pick = 0.05, then 5% of all our experiments will end in rejecting H 0, even though H 0 is true every time. So we should choose a lower value of . In this case, our result isn’t really “statistically significant.” We need a bigger sample!

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