Statistical Inference: One- Sample Confidence Interval Chapter 11 Statistical Inference: One- Sample Confidence Interval I Criticisms of Null Hypothesis Significance Testing  Does not indicate whether the effect is large or small

 A confidence interval for  is a segment on the  Answers the wrong question: Prob(D|H0). The correct question concerns Prob(H0|D).  Is a trivial exercise; all null hypotheses are false.  Turns a continuum of uncertainty into a reject-do- not reject decision. II Confidence Interval for   A confidence interval for  is a segment on the real number line such that that  has a high probability of lying on the segment.

Figure 1. Sampling distribution of t. If one t statistic is randomly sampled from this population of t’s, the probability is .95 that the obtained t will come from the interval from –t.05/2,  to t.05/2, .

1. From Figure 1, the following probability statement follows: 2. Replacing t with and using some algebra gives the following 100(1 – )% two-sided confidence interval for  L1 L2

3. L1 and L2 denote, respectively, the lower and upper endpoints of the open confidence interval for . 4. A researcher can be 100(1 – )% confident that  is greater than L1 and less than L2. 5. The probability (1 – ) is called the confidence coefficient and is usually equal to (1 – .05 ) = .95.

1. Consider the following hypotheses for the 6. The assumptions associated with a confidence interval are the same as those for a one-sample t statistic. A. Computational Example: Two-Sided Interval 1. Consider the following hypotheses for the Idle-On-In College registration example: H0:  =0 H1:  ≠ 0

2. A two-sided 100(1 – .05) = 95% confidence interval for , where

3. The dean can be 100(1 – .05) = 95% confident that  is greater than 2.78 and less than 3.02. 4. The dean can be even more confident that  lies in the interval from L1 to L2 by computing a 100(1 – .01) = 99% confidence interval.

5. A two-sided 100(1 – .01) = 99% confidence interval for , where t.01/2, 26 = 2.779, is given by

6. Graphs of the two confidence intervals 95% confidence interval for  99% confidence interval for 

B. More On the Interpretation of Confidence Intervals 7. As the dean’s confidence that she has captured  increases, so does the size of the interval from L1 to L2. B. More On the Interpretation of Confidence Intervals C. Computational Example: One-Sided Interval 1. Suppose that one-tailed hypotheses, H0:  ≥0 and H1:  <0, reflect the dean’s hunch about the new registration procedure.

2. A one-sided 100(1 – .05) = 95% confidence interval for , where

3. Comparison of one- and two-sided confidence intervals One-sided 95% confidence interval for  Two-sided 95% confidence interval for 

Over Hypothesis Testing D. Advantages of Confidence Interval Estimation Over Hypothesis Testing 1. Hypothesis testing is not very informative. A confidence interval narrows the range of possible values for . 2. Confidence intervals can be used to test all null hypotheses such as H0:  =0. Any 0 that lies outside of the confidence interval corresponds to a rejectable null hypothesis.

3. A sample mean and confidence interval provide an estimate of the population parameter and a range of values—the error variation—qualifying the estimate. 4. A 100(1 – )% confident interval for  contains all of the values of 0 for which the null hypothesis would not be rejected.

III Practical Significance A. Estimator of Cohen’s d 1. Hedges’s g for the registration example 2. Interpretation of g   

3. Computation of g from t statistics in research reports 4. For the registration example, t = 3.449 and n = 27