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Interval Estimation for Means

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Interval Estimation for Means. Notes of STAT6205 by Dr. Fan. Overview. Sections 6.2 and 6.3 Introduction to interval estimation Confidence Intervals for One mean General construction of a confidence interval Confidence Intervals for difference of two means Pair Samples.

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Interval Estimation for Means

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  1. Interval Estimationfor Means Notes of STAT6205 by Dr. Fan

  2. Overview • Sections 6.2 and 6.3 • Introduction to interval estimation • Confidence Intervals for One mean • General construction of a confidence interval • Confidence Intervals for difference of two means • Pair Samples 6205-ch6

  3. Interval Estimation 6205-ch6

  4. Confidence vs. Probability The selection of sample is random. But nothing is random after we take the sample! 6205-ch6

  5. (Symmetric) Confidence Interval • A k% confidence interval (C.I.) for a parameter is an interval of values computed from sample data that includes the parameter k% of time: Point estimate +multiplier x standard error • K% of time = k% of all possible samples 6205-ch6

  6. Estimation of One Mean m When the population distribution is normal Case 1: the SD s is known  Z interval Case 2: the SD is s unknown  t interval 6205-ch6

  7. Estimation of One Mean m When the population distribution is not normal but sample size is larger (n> = 30) Case 1: the SD s is known Z interval Case 2: the SD is s unknown Z interval, replacing s by s. 6205-ch6

  8. 6205-ch6

  9. Examples/Problems 6205-ch6

  10. Examples/Problems • Example 1: We would like to construct a 95% CI for the true mean weight of a newborn baby. Suppose the weight of a newborn baby follows a normal distribution. Given a random sample of 20 babies, with the sample mean of 8.5 lbs and sample s.d. of 3 lbs, construct such a interval estimate. 6205-ch6

  11. Can CI be Asymmetric? • Endpoints can be unequal distance from the estimate • Can be one-sided interval Example: Repeat Example 1 but find its one-sided interval (lower tailed). • Why symmetric intervals are the best when dealing with the normal or t distribution unless otherwise stated? 6205-ch6

  12. How to Construct Good CIs • Wish to get a short interval with high degree of confidence Tradeoff: • The wider the interval, the less precise it is • The wider the interval, the more confidence that it contains the true parameter value. Best CI: For any given confidence level, it has the shortest interval. 6205-ch6

  13. Difference of Two Means When: Two independent random samples from two normal populations Case 1: variances are known Z interval Case 2: variances are unknown without equal variance assumption Approximate t interval with equal variance assumption Pooled t interval 6205-ch6

  14. Difference of Two Means When: 2 independent random samples from two non-normal populations but large samples (n1, n2 >= 30) Case 1: variances are known Z interval Case 2: variances are unknown Z interval, replacing siby Si. 6205-ch6

  15. 6205-ch6

  16. Examples/Problems • Example 2: Do basketball players have bigger feet than football players? • Example 3: To compare the performance of two sections, a test was given to both sections. • From an estimation point of view (for variances), why is the pooled method preferred? • How to check the assumption of equal variance? 6205-ch6

  17. Example 6205-ch6

  18. Example 6205-ch6

  19. Paired Samples 6205-ch6

  20. 6205-ch6

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