Estimation and Confidence Intervals

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2 Estimation Estimation a statistical procedure in which a sample statistic is used to estimate the value of an unknown population parameter. Two types of estimation are: Point estimation Interval estimation The goal of estimation is similar to that of hypothesis testing, which is to learn more about the value of a mean or mean difference in a population of interest

3 Estimation Point estimation is a statistical procedure that uses a sample statistic to estimate the a population parameter For example, the sample mean is an estimate of the unknown population mean µ Advantage it is an unbiased estimator, that is, the sample mean will equal the population mean on average Disadvantage have no way of knowing for sure whether a sample mean equals the population mean For this reason, researchers often report the sample mean (a point estimate) and an interval within which a population mean is likely to be contained (an interval estimate)

4 Estimation Interval estimation is a statistical procedure in which a sample of data is used to find the interval or range of possible values within which a population parameter is likely to be contained An interval estimate is stated within a given level of confidence Confidence interval (CI) interval or range of possible values within which an unknown population parameter is likely to be contained Level of confidence probability or likelihood that an interval estimate will contain an unknown population parameter (e.g., population mean)

5 Estimation and Confidence Intervals

6 Confidence Intervals for the Population ean Steps for computing a confidence interval: 1. Compute the sample mean (or difference of sample means) and the standard error (or standard error estimate) 2. Choose the level of confidence (e.g., 95%) and look up the corresponding critical t (or z) values in the t (or z) distribution table. 3. Compute the confidence interval by inverting the t (or z) transformation (this should be familiar our earlier exercises computing x from z) E.g., for a z-statistic: CI z 1 z n

7 Back to Our Test Scores Example Estimation and Confidence Intervals We sample 5 (i.e., n=5) students from Dr. s class, administer the test and find that their average score is What is our 95% confidence interval estimate (CI.95 ) for the population mean of Dr. s class? As with a two-tailed test, when evaluating CI s, we want half of the residual probability in each tail I.e., for CI.95 we want (1-0.95)/2 = in each tail

8 Computing CI s for the z-statistic Estimation and Confidence Intervals n CI z z 0.05 n

9 z Upper-Tail Probabilities

10 z Upper-Tail Probabilities

11 Computing CI s for the z-statistic Estimation and Confidence Intervals n z CI z z 0.05 n 1.96(7.0) CI [68.86,81.14]

12 Computing CI s for the z-statistic Estimation and Confidence Intervals 95% CI CI.95 (µ) (interval estimate of µ) z µ 0 (point estimate of µ)

13 Hypothesis Testing with z-statistic Estimation and Confidence Intervals z µ 0 (from H 0 )

14 Computing CI s for the One-Sample t-statistic For the moon illusion example: s n CI t s t 0.05 n s

15 t-distribution Table t One-tailed test α/2 -t t Two-tailed test α α/2 Level of significance for one-tailed test Level of significance for two-tailed test df

16 t-distribution Table t One-tailed test α/2 -t t Two-tailed test α α/2 Level of significance for one-tailed test Level of significance for two-tailed test df

17 Computing CI s for the One-Sample t-statistic For the moon illusion example: s n t CI t s t0.05s n 2.262( 0.341) [1.22,1. 71]

18 Computing CI s for the One-Sample t-statistic For the moon illusion example: s n t CI t s t0.05s n 2.262( 0.341) [1.22,1. 71]

19 (from H ) 0 0 t0.05s t0.05s

20 For the anorexia-family therapy example: s CI t s Estimation and Confidence Intervals Computing CI s for the independent-samples t-statistic n1 17 Compute Pooled Variance: Estimate Standard Error: df1s1 df2s2 s s sp sp p df s1 2 1 df2 n1 n2 n

21 t-distribution Table t One-tailed test α/2 -t t Two-tailed test α α/2 Level of significance for one-tailed test Level of significance for two-tailed test df

22 Computing CI s for the independent-samples t-statistic For the Anorexia / Family Therapy example: s 1 n 1 s n t s 1 2 CI CI ( 0.45) 2.021(2.40) [2.86,12.56] s t

23 t s t s