Confidence interval & probability statements
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This document discusses confidence intervals and probability. It defines confidence intervals as a range of values that provide more information than a point estimate by taking into account variability between samples. The document provides examples of how to calculate 95% confidence intervals for a proportion, mean, odds ratio, and relative risk using sample data and the appropriate formulas. It explains that confidence intervals convey the level of uncertainty associated with point estimates and allow estimation of how close a sample statistic is to the unknown population parameter.
Confidence interval & probability statements
- 1. Confidence Interval & Probability Dr Zahid Khan SENIOR LECTURER KING FAISAL UNIVERSITY
- 2. 2 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
- 3. Point and Interval Estimates 3 A point estimate is a single number, a confidence interval provides additional information about variability Lower Confidence Limit Point Estimate Width of confidence interval Upper Confidence Limit
- 4. 4 Point Estimates We can estimate a Population Parameter … with a Sample Statistic (a Point Estimate) Mean μ x Proportion p p
- 5. Confidence 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
- 6. Estimation Process Random Sample Population (mean, μ, is unknown) Sample Mean x = 50 I am 95% confident that μ is between 40 & 60.
- 7. Confidence interval endpoints Upper and lower confidence limits for the population proportion are calculated with the formula p z/2 p(1p) n where z is the standard normal value for the level of confidence desired p is the sample proportion n is the sample size
- 8. 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
- 9. Example A random sample of 100 people shows that 25 are lefthanded. Form a 95% confidence interval for the true proportion of left-handers. .25 1. p 25/100 (1 p .0433 2. S p )/n .25(.75)/n p 3. .251.96 (.0433) 0.16510.3349 .....
- 10. Interpretation We are 95% confident that the true percentage of lefthanders 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.
- 11. 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
- 12. 95% CI for Mean μ+ 1.96 * SE SE= SD²/n SE difference = SD²/n1 + SD²/n2
- 13. 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)
- 14. 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.
- 15. CI for Relative Risk Dead Alive Total Placebo 21 110 131 Isoniazid 11 121 132
- 16. 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