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

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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