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Hypothesis Tests for Means

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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Hypothesis Tests for Means

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  1. 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

  2. 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 

  3. 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 notsignificantly different from 0. • IF NO – then we REJECT the hypothesis that  = 0. We say that x IS significantly different from 0.

  4. 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.

  5. Hypothesis Test for  • Cookbook using rejection regions • 1. Choose hypotheses – H0 and HA. • 2. Define a test statistic. • 3. Predict the distribution of the test statistic, • assuming that H0 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 H0.

  6. Hypothesis Test for  • Cookbook using rejection regions • 1. Choose hypotheses – H0 and HA. • 2. Define a test statistic. • 3. Predict the distribution of the test statistic, • assuming that H0 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 H0.

  7. Choose hypotheses • Two-sided test: • H0:  = 0 HA: 0 • One-sided tests: • H0:  = 0 HA:  > 0 • or • H0:  = 0 HA:  < 0 • Working rule: Always use two-sided tests.

  8. Hypothesis Test for  • Cookbook using rejection regions • 1. Choose hypotheses – H0 and HA. • 2. Define a test statistic. • 3. Predict the distribution of the test statistic, • assuming that H0 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 H0.

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

  10. Hypothesis Test for  • Cookbook using rejection regions • 1. Choose hypotheses – H0 and HA. • 2. Define a test statistic. • 3. Predict the distribution of the test statistic, • assuming that H0 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 H0.

  11. Distribution of the test statistic • ASSUME H0 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.

  12. Hypothesis Test for  • Cookbook using rejection regions • 1. Choose hypotheses – H0 and HA. • 2. Define a test statistic. • 3. Predict the distribution of the test statistic, • assuming that H0 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 H0.

  13. Rejection region consists of two parts, each with probability /2. z*/2 - z*/2 (Standard normal case) • The rejection region is a range (or double-range) of values of the test statistic that are • (a) UNLIKELY if H0 is true • (b) roughly consistent with the alternative HA. • The rejection region should have probability  (given H0). • Two-sided case:

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

  15. Chance of a Type I error • Note: • IF H0 is actually true, then there is still a probability of  that you will reject the null hypothesis. z*/2 - z*/2

  16. Chance of a Type I error • There are two possible bad results: • TYPE I ERROR (“act of commission”) – reject H0, when H0 is actually true. • The probability of a Type I error is  • (given that H0 is true) • TYPE II ERROR (“act of omission”) – don’t reject H0, when H0 is actually false. • The probability of a Type II error depends • on the actual value of 

  17. Hypothesis Test for  • Cookbook using rejection regions • 1. Choose hypotheses – H0 and HA. • 2. Define a test statistic. • 3. Predict the distribution of the test statistic, • assuming that H0 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 H0.

  18. 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?

  19. 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 HA: 0 this means, what fraction are further from zero than the value you obtained? • If HA:  > 0 this means, what fraction are more than the value you obtained? • If HA:  < 0 this means, what fraction are less than the value you obtained?

  20. tail: 0.0107 z=2.30 Determine p-value • Example: • Do a test of H0:  = 0 vs. HA:   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

  21. Determine p-value • Keep it simple? • p-value = • (for 1-sided test with z) = NORMSDIST ( 1 - |z| ) • (for 2-sided test with z) = 2 × NORMSDIST(1-|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

  22. Determine p-value • The p-value is the border between ’s for which • we reject H0 and ’s for which we do not • reject H0. • 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 H0 if p <  • Do not reject if p > .

  23. 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): H0 :  = 183.5 • (and this is one of those cases where  = 105.6547 comes with the null hypothesis) • Alternative: HA :   183.5

  24. Example: 1969 Draft Lottery • H0 :  = 183.5 HA :   183.5 • 0 = 183.5 •  = 105.6547 • Experiment: n = 184, x = _________ • Test statistic: • p-value: • Conclusion: REJECT H0 (even at 1% significance level) 160.92 = - 2.898 0.00375

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

  26. Hypothesis tests for proportions • Test statistic: • (Minor subtlety: The distribution of the test statistic is based on H0, so we use p0 in the formula for SE. This is different from what we do in confidence intervals, but not by much.)

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

  28. Another example • Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant? • Choose: • H0: p = 0.50 HA: p  0.50

  29. Another example • Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant? • Choose: • H0: p = 0.50 HA: p  0.50 • Conditions? OK.

  30. Another example • Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant? • Choose: • H0: p = 0.50 HA: p  0.50 • Conditions? OK. • Distribution of p^, given H0: • Normal, mean 0.50, SD=0.005

  31. 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 ?

  32. 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.

  33. Result of test • Is a P-value of 0.045 good enough to reject H0?

  34. Result of test • Is a P-value of 0.045 good enough to reject H0? • If we choose  = 0.05, then yes. But that’s a very mild test for such an extraordinary claim.

  35. Result of test • Is a P-value of 0.045 good enough to reject H0? • 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 H0, even though H0 is true every time.

  36. Result of test • Is a P-value of 0.045 good enough to reject H0? • 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 H0, even though H0 is true every time. • So we should choose a lower value of . In this case, our result isn’t really “statistically significant.”

  37. Result of test • Is a P-value of 0.045 good enough to reject H0? • 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 H0, even though H0 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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