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Interval Estimation for Means

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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