Confidence Interval
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1 iid We observed !
What is the true value of p , in the experiment ? Is exact value of ? No
for large 5
Plausible range that will be in ? (0, 1) : true but no meaning A reasonable way is . How to take ? when is large
95% confidence interval (CI) 95% range : solve w.r.t p for large n 95% confidence interval (CI)
95% CI of p : 9
95% CI of 90% CI of
> 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 = 127.449, df = 1, p-value < 2.2e-16 alternative hypothesis: true p is not equal to 0.5 95 percent confidence interval: 0.6489166 0.7076898 sample estimates: p 0.679 > prop.test(head,total,conf.level=0.9) 90 percent confidence interval: 0.6537527 0.7032614
is observed ! What is the plausible range of ?
is observed !
-1 1 2 3
is observed ! What is the plausible range of ?
95% range of : 95% CI of :
95% 1.96
95% -1.96 1.96 Range
Confidence Interval
A 0.75 0.05 B 0.1 0.5 C A B C likelihood
is observed. What is ?
distribution & likelihood p x 6 5 0.0 1 1.0 distribution & likelihood
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
90% 1.645 Range & Confidence Interval
25
95% range : 95% CI of : 26
When we know 90% CI of : 27
When we know 100(1- )% CI of : 28
When we don’t know ? 29
30
31
32
33
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
4.27, 7.74, 5.09, 4.25, 9.80, 7.71, 9.81, 6.63, 5.01 95% CI of ?
> 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: 5.009511 8.392711 sample estimates: mean of x 6.701111 > t.test(x,conf.level=0.9) 90 percent confidence interval: 5.337015 8.065207 4.27, 7.74, 5.09, 4.25, 9.80, 7.71, 9.81, 6.63, 5.01
95% CI of ?
★ unknown known
★ 95% CI of ?
★
Thank you !!