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Unformatted text preview: IV. Inference.
A. Confidence Intervals for Population Mean. (Chapter 8) 1. Point and Interval Estimation. 2. Confidence Intervals – σ known. 3. Margin of Error and Sample Size. 4. Confidence Intervals – σ unknown. Combine point estimation and the point sampling sampling distribution. 4. Confidence interval - σx is not known. • We don’t know μ and we don’t know σ. • Estimate both μ and σ. • More estimation, added uncertainty. • Use the t-distribution – it has fatter tails and gives wider confidence intervals. x − tα / 2 ⋅ sx n x x + tα / 2 ⋅ sx n 4. Confidence interval - σx is not known. t-distribution: standardized distribution – centered at 0. Looks a lot like the Z-distribution, but… • Shape depends on degrees of freedom: df = (n – 1) • As (n – 1) → ∞ , the t-distribution converges to a zdistribution. 1 4. Confidence interval - σx is not known. The mean and standard deviation for voltages of power packs labeled as 12 volts for a sample of 8 are as follows: x = 11.14 v and s = 0.3 v. . Create 90% confidence interval estimate for the Create a 90% confidence interval estimate for the population population mean. PRS 3: You wish to create a 90% confidence interval estimate of the mean textbook costs for ResEc 211 students. Using your personal sample of 16 observations, you estimated the population standard deviation standard deviation (σ). Report your estimate of σ – use two decimal places. sx = ∑ x − (∑ x)
2 2 n n −1 4. Confidence interval - σx is not known. Use Use your sample of n = 16 commute distances to compute a 90% confidence interval estimate of the true population mean commute distance.
Lower Limit Point Estimate Upper Limit x ± tα / 2 ⋅ sx n 2 B. Hypothesis Tests of Population Mean 1. Basics of Hypothesis Testing
Null Hypothesis • A statement about the true value of the population parameter, μx. • Specific value to be tested – need a number. • Notation: 1. Basics of Hypothesis Testing Alternative Hypothesis • Values of μx , if the null is not true. • The hypothesis we’re really interested in – the research hypothesis. • Notation: 1. Basics of Hypothesis Testing Complete Hypothesis - examples • Machine filling 20 oz. bottles. Concern: bottles do not contain 20 oz. We don’t want the machine to put in too much or too little: in too much or too little: 3 1. Basics of Hypothesis Testing Complete Hypothesis - examples • Textbook Expenditures – concern on campus that textbook costs have increased. Fall 2008 semester mean expenditures by students in ResEc 211 were $313: 1. Basics of Hypothesis Testing Complete Hypothesis - examples • Time on Task – concern is that students spend too little “time on task.” Testing suggests that the time to carefully read and respond to OWL question number 8635 on the margin of error is 32 seconds : A sample of 791 ResEc 211 student responses resulted in an average of 27 seconds with a sample standard deviations of 83 seconds. 4 ...
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This note was uploaded on 12/08/2011 for the course ECON 211 taught by Professor Daniellass during the Spring '11 term at UMass (Amherst).
- Spring '11