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Confidence Interval & Probability Statements
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Confidence Interval &ProbabilityDr Zahid KhanSENIOR LECTURER KING FAISAL UNIVERSITY
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Confidence Intervals How much uncertainty is associated
with a point estimate of a population parameter?
An interval estimate provides more information about a population characteristic than does a point estimate
Such interval estimates are called confidence intervals
3Point and Interval Estimates
A point estimate is a single number, a confidence interval provides additional
information about variability
Point Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Width of confidence interval
4
Point Estimates
We can estimate a Population Parameter …
with a SampleStatistic
(a Point Estimate)
Mean
Proportion pp
xμ
5Confidence Interval Estimate
An interval gives a range of values: Takes into consideration variation in
sample statistics from sample to sample
Based on observation from 1 sample Gives information about closeness to
unknown population parameters Stated in terms of level of confidence
Never 100% sure
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Estimation Process
(mean, μ, is unknown)
Population
Random Sample
Mean x = 50
Sample
I am 95% confident that μ is between 40 & 60.
7Confidence interval endpoints Upper and lower confidence limits for the
population proportion are calculated with the formula
where z is the standard normal value for the level of confidence
desired
p is the sample proportion
n is the sample size
n
)p(pzp /2
1
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Example A random sample of 100 people shows that 25 are left-handed.
Form a 95% confidence interval for the true proportion of left-handers
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Example A random sample of 100 people shows that 25 are left-
handed. Form a 95% confidence interval for the true proportion of left-handers.
1.
2.
3.
.0433 .25(.75)/n)/np(1pS
.2525/100 p
p
0.3349 . . . . . 0.1651
(.0433) 1.96 .25
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Interpretation We are 95% confident that the true percentage of left-
handers in the population is between
16.51% and 33.49%.
Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.
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Changing the sample size Increases in the sample size reduce the width of the confidence
interval.
Example: If the sample size in the above example is doubled to
200, and if 50 are left-handed in the sample, then the interval is still centered at .25, but the width shrinks to
.19 …… .31
95% CI for Mean
μ+ 1.96 * SE SE= SD²/n SE difference = SD²/n1 + SD²/n2
CI for Odds Ratio
CASES
Appendicitis Surgical ( Not appendicitis)
Females 73(a) 363(b)
Males 47(c ) 277(d)
Total 120 640
OR = ad/bc
95% CI OR = log OR + 1.96 * SE (Log OR)
CI for OR
SE ( loge OR) = 1/a + 1/b + 1/c + 1/d
= 1/73 + 1/363 + 1/47 + 1/277 = 0.203 Loge of the Odds Ratio is 0.170.
95% CI = 0.170 – 1.96 * 0.203 to 0.170 * 1.96 * 0.203
Loge OR = -0.228 to 0.578
Now by taking antilog ex we get 0.80 to 1.77 for 0.228 and 0.578 respectively.
CI for Relative Risk
Dead Alive Total
Placebo 21 110 131
Isoniazid 11 121 132
CI for Relative Risk
SE ( LogRR) = 1/a – 1/a+b + 1/c – 1/c+d
SE (LogRR) = 1/21-1/131 + 1/11 – 1/132 = 0.351 RR = a/ a+b / c/ c+d = 0.52
LogRR = Log 0.52 = - 0.654
95% CI = -0.654 -1.96 * 0.351 , -0.654 +1.96 * 0.351 = -1.42, 0.040 so by taking anti log we have 95% CI = 0.242, 1.04