Confidence interval & probability statements
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Confidence interval & probability statements

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Confidence Interval & Probability Statements

Confidence Interval & Probability Statements

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Confidence interval & probability statements Confidence interval & probability statements Presentation Transcript

  • Confidence Interval & Probability Dr Zahid Khan SENIOR LECTURER KING FAISAL UNIVERSITY
  • 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
  • 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 Point Estimates We can estimate a Population Parameter … with a Sample Statistic (a Point Estimate) Mean μ x Proportion p p
  • 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
  • Estimation Process Random Sample Population (mean, μ, is unknown) Sample Mean x = 50 I am 95% confident that μ is between 40 & 60.
  • Confidence interval endpoints  Upper and lower confidence limits for the population proportion are calculated with the formula p  z/2  p(1p) n where  z is the standard normal value for the level of confidence desired  p is the sample proportion  n is the sample size
  • 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
  • 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 .....
  • 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.
  • 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