Interval Estimation for Means Notes of STAT6205 by Dr. Fan.

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Presentation transcript:

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 6205-ch62

Interval Estimation 6205-ch63

Confidence vs. Probability 6205-ch64 The selection of sample is random. But nothing is random after we take the sample!

(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-ch65

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-ch66

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 ch67

8

Examples/Problems 9

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 ch610

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-ch611

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 ch612

Difference of Two Means When: Two independent random samples from two norma l 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-ch613

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 s i by Si ch614

6205-ch615

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-ch616

Example 6205-ch617

Example 6205-ch618

Paired Samples 6205-ch619

6205-ch620