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Confidence Interval
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1 iid We observed !
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What is the true value of p , in the experiment ?
Is exact value of ? No
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for large 5
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Plausible range that will be in ?
(0, 1) : true but no meaning A reasonable way is How to take ? when is large
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95% confidence interval (CI)
95% range : solve w.r.t p for large n 95% confidence interval (CI)
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95% CI of p : 9
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95% CI of 90% CI of
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> head<- 679 ; total<-1000
> prop.test(head,total) 1-sample proportions test with continuity correction data: head out of total, null probability 0.5 X-squared = , df = 1, p-value < 2.2e-16 alternative hypothesis: true p is not equal to 0.5 95 percent confidence interval: sample estimates: p 0.679 > prop.test(head,total,conf.level=0.9) 90 percent confidence interval:
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is observed ! What is the plausible range of ?
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is observed !
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-1 1 2 3
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is observed ! What is the plausible range of ?
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95% range of : 95% CI of :
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95% 1.96
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95% -1.96 1.96 Range
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Confidence Interval
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A 0.75 0.05 B 0.1 0.5 C A B C likelihood
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is observed. What is ?
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distribution & likelihood
p x 6 5 0.0 1 1.0 distribution & likelihood
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0.0 0.2 0.4 0.6 0.8 1.0 0.1 0.3 p likelihood 0.133 0.587 0.15 1 2 3 4 5 6 0.00 0.10 0.20 0.30
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90% 1.645 Range & Confidence Interval
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95% range : 95% CI of : 26
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When we know 90% CI of : 27
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When we know 100( )% CI of : 28
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When we don’t know ? 29
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0.3 1.0 0.8 0.2 0.6 0.4 0.1 0.2 0.0 0.0 -15 -10 -5 5 10 15 -15 -10 -5 5 10 15 1 3 8 15 21 23 ∞ 0.10 3.078 1.638 1.397 1.341 1.323 1.319 1.282 0.05 6.314 2.353 1.860 1.753 1.721 1.714 1.645 0.025 12.706 3.182 2.306 2.131 2.080 2.069 1.96
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4.27, , , 4.25, , , , , 95% CI of ?
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> x<-c(4.27,7.74,5.09,4.25,9.80,7.71,9.81,6.63,5.01) > > t.test(x) One Sample t-test data: x t = 9.135, df = 8, p-value = 1.661e-05 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: sample estimates: mean of x > t.test(x,conf.level=0.9) 90 percent confidence interval: 4.27, , , 4.25, , , , ,
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95% CI of ?
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★ unknown known
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★ 95% CI of ?
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★
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Thank you !!
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