Interpretation of results through confidence intervals

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Alexander Wright
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1 Interpretation of results through confidence intervals

2 Hypothesis tests Confidence intervals Hypothesis Test Reject H 0 : μ = μ 0 Confidence Intervals μ 0 is not in confidence interval μ 0 P(observed statistic μ = μ 0 ) < α P(observed conf. int. μ = μ 0 ) < α P(observed data μ = μ 0 ) < α

3 Hypothesis tests Confidence intervals If μ 0 is outside the (1- α)% confidence interval, then H 0 : μ = μ 0 will be rejected at significance level α. μ 0 Reject H 0 : μ = μ 0 If μ 0 is inside the (1- α)% confidence interval, then H 0 : μ = μ 0 will not be rejected at significance level α. μ 0 Do not reject H 0 : μ = μ 0

Example: change in facial height μ = mean change in facial height n = 84, X = 1.7, s = 6.8 mm. The p-value for the test of H 0 : μ = 0 is p = 0.024. Reject H 0 at α = 0.

4 Example: change in facial height μ = mean change in facial height n = 84, X = 1.7, s = 6.8 mm. The p-value for the test of H 0 : μ = 0 is p = Reject H 0 at α = 0.05 Do not reject H 0 at α = % conf. int. for μ: = (-3.2, -0.2) does not contain 0 99% conf. int. for μ: = (-3.7, +0.3) contains 0

5 Example: chewing gum data μ = mean change DMFS n = 25, X = 0.72, s = % conf. int. for μ: = (-2.9, 1.5) We would reject H 0 : μ = -3.0 or H 0 : μ = 2.0 We would not reject H 0 : μ = 0

6 Example: chewing gum data μ = mean change DMFS n = 25, X = 0.72, s = % conf. int. for μ: = (-2.9, 1.5) We would also not reject H 0 :μ = -2.0 or H 0 :μ = +1.0 Thus, even though we did not reject H 0 : μ = 0, it would be incorrect to say, we accept μ = 0. Best to say only, we don t have enough evidence to reject.

7 Confidence intervals help interpret negative results The confidence interval divides all possible values for μ into: Those that our data appear to contradict, and we can safely say are not true (reject) And those that our data do not contradict (do not reject) not true! maybe true? not true! confidence interval The study has good evidence regarding values outside of the confidence interval. The study has good evidence to say the values outside the confidence interval are not true.

8 Confidence Intervals: The values outside of the interval are the real evidence of the study

9 Hypothesis test for proportions

10 One-sample test for proportions One sample test for proportions is used to make inference about a proportion, p, in the population Based on the statistic p p 0 Z = p 0 1 p 0 n If Z is large (far from zero), it gives evidence that p p 0. Z will follow a standard Normal distribution if: n large enough (np(1-p) > 5), and p = p 0. Can test H 0 :p=p 0 by comparing Z to a standard Normal.

11 Example: Presidential Election Poll Presidential preference poll (completed 7/15/2016) Trump Clinton Total Do these data provide evidence that Trump had a lead in the population sampled? To answer this question we can test the hypothesis H 0 : p < 0.5 vs H 1 : p > 0.5 where p is proportion in population favoring Trump.

12 Example: Presidential Election Poll Test the hypothesis H 0 : p < 0.5 vs H 1 : p > 0.5 Note the form of the hypotheses differs from those previously introduced. Rejecting H 0 will indicate evidence that Trump has a lead (p > 0.5) If we had used H 0 : p = 0.5 vs H 1 : p 0.5, then rejecting H 0 would indicate only that the candidates were not tied. This formulation is an example of a one-sided test

13 Example: Presidential Election Poll Test the hypothesis Compute the Z statistic p = Z = H 0 : p < 0.5 vs H 1 : p > = p p 0 p 0 1 p 0 n = (1 0.5) 780 = Note that for p 0 we use the border probability, 0.5. This would be the worst case in terms of trying to disprove H 0. P-value =?

14 Example: Presidential Election Poll Test the hypothesis H 0 : p < 0.5 vs H 1 : p > 0.5 Z = p n = For a one-sided test the p-value is one sided. The p-value is the probability of Z = or more extreme Extreme meaning more unlikely if H 0 is true. Since H 0 specifies p s less than 0.5, the p-value will be: P-value = P(Z > 0.726) = 0.234

15 Example: Presidential Election Poll If one cannot compute the p-value easily, one can also compare the Z statistic to a critical value to see if the p-value is less than or greater than α. In the case of a one-sided test at significance level α: If H 1 : p > p 0, then reject H 0 if Z > Z 1-α If H 1 : p < p 0, then reject H 0 if Z < Z α Note that we do not halve the α for one-sided tests. If α = 0.05, we would compare Z = to Z 0.95 = Since < 1.645, we do not reject H 0.

16 Final notes on Presidential Election Poll example Though we did not reject H 0, this does NOT necessarily indicate that H 0 is true. The 95% confidence interval for p is (47.8%, 54.8%) This indicates that there is NOT strong evidence of either candidate having the majority. General notes Tests for proportions do not have to be one-sided. Two-sided tests for proportions can also be formulated. One-sided tests can be used for comparisons other than proportions (e.g. one can use them with t-tests).

Hypothesis Test-Confidence Interval connection

Hypothesis Test-Confidence Interval connection Hyothesis Test-Confidence Interval connection Hyothesis tests for mean Tell whether observed data are consistent with μ = μ. More secifically An hyothesis test with significance level α will reject the