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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
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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.
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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, .
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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
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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.
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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
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2. A two-sided 100(1 – .05) = 95% confidence
interval for , where
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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.
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5. A two-sided 100(1 – .01) = 99% confidence
interval for , where t.01/2, 26 = 2.779, is given by
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6. Graphs of the two confidence intervals
95% confidence interval for 99% confidence interval for
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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.
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2. A one-sided 100(1 – .05) = 95% confidence
interval for , where
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3. Comparison of one- and two-sided confidence
intervals One-sided 95% confidence interval for Two-sided 95% confidence interval for
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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.
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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.
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III Practical Significance
A. Estimator of Cohen’s d 1. Hedges’s g for the registration example 2. Interpretation of g
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3. Computation of g from t statistics in research
reports 4. For the registration example, t = and n = 27
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