Chapter 3 Confidence Interval

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Chapter 3 Confidence Interval

  1. 1. MEDICAL STATISTICS <ul><li>CHAPTER-3 </li></ul>
  2. 2. <ul><li>Chapter 3 </li></ul><ul><li>Sampling Error </li></ul><ul><li>and Confidence Interval </li></ul>
  3. 3. For several samples from the same population <ul><li>Usually </li></ul><ul><li>the sample means are not equal to the population mean </li></ul><ul><li>the sample means are different one another </li></ul><ul><li>----sampling error </li></ul>
  4. 4. 3.1 The Distribution of Sample Mean <ul><li>3.1.1 Distribution of sample mean from a population of normal distribution </li></ul><ul><li>Experiment 3.1 Sampling from a normal distribution. </li></ul><ul><li>Assume the red cell counts of healthy males follow a normal distribution </li></ul><ul><li>100 samples are drawn, </li></ul><ul><li>The sample means are showed in the second column of Table 3.1. </li></ul>
  5. 6. Features of sample mean as a random variable <ul><li>Any of the sample means is not necessary equal to the population mean; </li></ul><ul><li>(2) The differences exist among the sample means; </li></ul>
  6. 7. (3) The distribution of sample means follows certain rule that more in center, less in two ends and symmetry around the center. (4) The range of variation for the sample mean is much narrower than that of the initial variable.
  7. 8. If the random samples with n individuals are drawn from a normal distribution , then the sample mean follows a normal distribution (3.1) , then the sample mean follows a normal distribution (3.1)
  8. 9. <ul><li>(5) The range of variation for the sample means tends to be narrow with the increase of sample sizes. </li></ul><ul><li>: Standard deviation of the initial variable </li></ul><ul><li>: Standard deviation of the sample mean </li></ul><ul><li>---- standard error of sample mean </li></ul><ul><li>or standard error . </li></ul><ul><li>For sample, </li></ul>
  9. 10. 3.1.2 Distribution of sample mean from a population with non-normal distribution <ul><li>Experiment 3.2 Sampling from positive skew distribution </li></ul>
  10. 11. <ul><li>The distribution of sample means tends to be symmetric with the increase of sample size; </li></ul><ul><li>when n =30, it looks similar to normal distribution. </li></ul><ul><li>(2) The range of variation for the sample means also tends to be narrow with the increase of sample sizes. </li></ul>
  11. 12. <ul><li>Experiment 3.3 Sampling from an asymmetric hook-like distribution </li></ul>
  12. 13. <ul><li>For the population with a non-normal distribution , </li></ul><ul><li>although the distribution of sample means is not a </li></ul><ul><li>normal distribution, it will be similar to a normal </li></ul><ul><li>distribution when sample size is big (say, </li></ul><ul><li>approximately, we still have </li></ul>
  13. 14. 3.2 t Distribution <ul><li>3.2.1 Standard t deviate </li></ul><ul><li>When </li></ul>W.S. Gosett (1908) explored its distribution
  14. 15. 3.2.2 The probability density and critical values of t distribution
  15. 16. <ul><li>The two-side probabilities and corresponding critical </li></ul><ul><li>values of t distribution are given in the Table 5 of the </li></ul><ul><li>Appendix 2. </li></ul><ul><li>For instance, </li></ul><ul><li>When degrees of freedom is 20 , corresponding </li></ul><ul><li>to two-side probability 0.05 , the critical value of t </li></ul><ul><li>distribution </li></ul><ul><li>Corresponding to one-side probability 0.05 , the </li></ul><ul><li>critical value of t distribution </li></ul><ul><li>In general, </li></ul>
  16. 17. 3.3 The Confidence Interval for Population Mean of Normal Distribution Therefore, 95% of the sample means meet the inequality (but not all) For any sample, if we claim is located in such an interval, then in theory, we might be right about 95 times out of 100 times. and are unknown A sample is drawn, and ,
  17. 20. <ul><li>In general , given a random sample of the population, </li></ul><ul><li>if the sample size, sample mean and sample standard </li></ul><ul><li>deviation are denoted as ; , then </li></ul><ul><li>is called with confidence interval of the </li></ul><ul><li>population mean </li></ul><ul><li>: confidence level </li></ul><ul><li>: precision of the confidence interval </li></ul><ul><li>When sample size is big enough , </li></ul>
  18. 21. <ul><li>Example 3.1 Randomly select 20 cases from the patients </li></ul><ul><li>with certain kind of disease. The sample mean of blood </li></ul><ul><li>sedimentation (mm/h) ( 血沉 ) is 9.15, sample standard </li></ul><ul><li>deviation is 2.13. To estimate the 95% confidence interval </li></ul><ul><li>and 99% confidence interval (Assume the blood </li></ul><ul><li>sedimentation of this kind of disease follow a normal </li></ul><ul><li>distribution). </li></ul><ul><li>Question: If both of higher confidence level and better </li></ul><ul><li>precision are expected, What should we do? </li></ul>
  19. 22. 3.4 Confidence Interval for the Difference between Two Population Means unknown. Two samples with The confidence interval for ?
  20. 23. <ul><li>Since is unknown, it could be replaced by , </li></ul>
  21. 24. <ul><li>Example 3.2 Assume the red cell counts of healthy male </li></ul><ul><li>residents and healthy female residents of certain city </li></ul><ul><li>follow two normal distributions respectively </li></ul><ul><li>, , </li></ul><ul><li>95% CI for the difference between male and female? </li></ul>
  22. 25. 3.5 Confidence Intervals for Probability and the Difference between Two Probabilities <ul><li>3.5.1 Confidence interval for population probability </li></ul><ul><li>When sample size is small , given X and n the 95% and </li></ul><ul><li>99% confidence interval of can be obtained from Table </li></ul><ul><li>3 of Appendix 2. </li></ul><ul><li>When sample size is big enough , can be estimated </li></ul><ul><li>by normal approximation </li></ul>
  23. 26. Comparing to the confidence interval of
  24. 27. 3.5.2 Confidence intervals for two population probabilities
  25. 28. Comparing to the confidence interval of
  26. 29. <ul><li>Example 3.4 Comparison between two drugs. </li></ul>
  27. 30. 3.6 The Sample Size for Estimation of Confidence Interval <ul><li>3.6.1 Sample size for confidence interval of the mean of normal population </li></ul>3.6.1 Sample size for confidence interval of the mean of normal population Given (1) the confidence level (1-  ) (2) the half width of confidence interval δ (3) the estimate of the standard deviation s Let Replace with , approximately
  28. 31. <ul><li>Example 3.5 It is learnt from a pilot study that the </li></ul><ul><li>standard deviation of a biochemical index is about 10 </li></ul><ul><li>units. In order to have a 95% confidence interval of </li></ul><ul><li>the population mean, of which the half of the width </li></ul><ul><li>equals to 2.5 units. What is the sample size needed? </li></ul><ul><li>Since s =10, δ =2.5, ≈2, </li></ul>
  29. 32. 3.6.2 Sample size for confidence interval of the probability of binomial population <ul><li>Given (1) the confidence level (1-  ), </li></ul><ul><li>(2) the half width of confidence interval δ </li></ul><ul><li>(3) the estimate of frequency p </li></ul><ul><li>Let </li></ul><ul><li>This formula shows, the large sample size will be needed if </li></ul><ul><li>the population probability is close to 0.5 (big variation). </li></ul>
  30. 33. <ul><li>Example 3.6 It is learnt from a pilot study that the </li></ul><ul><li>probability of relapse in one year for a disease is about </li></ul><ul><li>10%. Now a survey is planed to further estimate the 95% </li></ul><ul><li>confidence interval for the probability of relapse in one </li></ul><ul><li>year, of which the half width is required with 3%. What </li></ul><ul><li>is the sample size needed? </li></ul><ul><li>Since p =10%, ≈2, </li></ul>
  31. 34. Summary <ul><li>1. Sampling error The sample means are not equal to the population mean; the sample means are different one another. </li></ul><ul><li>2. Distribution of sample mean If the random samples with n individuals are drawn from a normal distribution , then the sample mean follows a normal distribution. </li></ul><ul><li>If the random samples with n individuals are drawn from a non-normal distribution , although the distribution of sample means is not a normal distribution, it will be approximate to a normal distribution when sample size is big . </li></ul>
  32. 35. <ul><li>3. Confidence interval </li></ul><ul><li>When </li></ul><ul><li>Given a random sample of </li></ul><ul><li>if the sample size, sample mean and sample </li></ul><ul><li>standard deviation are denoted as , </li></ul><ul><li>then the confidence interval of the </li></ul><ul><li>population mean is </li></ul>
  33. 36. <ul><li>Given a random sample of , </li></ul><ul><li>then the confidence interval of the population </li></ul><ul><li>probability i s </li></ul><ul><li>Given two random samples of </li></ul><ul><li>and then the confidence interval of </li></ul><ul><li>the difference is </li></ul>
  34. 37. <ul><li>Given two random samples of and then the confidence interval of </li></ul><ul><li>the difference is </li></ul><ul><li>4. Sample size </li></ul><ul><li>Sample size for confidence interval of the mean of normal population </li></ul><ul><li>Sample size for confidence interval of the probability of binomial population </li></ul>
  35. 38. <ul><li>Thanks </li></ul>

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