Lecture 29 - 211 Fall 2009

Lecture 29 - 211 Fall 2009 - IV. Inference. A. Confidence...

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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. A. Confidence Intervals for Population Mean. 1. Point Estimate: a single numeric value – our best guess about µx 2. Confidence Interval Estimate • Combine: Point estimate – sample mean, a single numeric value Standard error – summary of variation for the sample mean • Choose a level of confidence: (1 – α) 2. Confidence Intervals – σX known Lower Limit Point Estimate Upper Limit x − zα / 2 ⋅ σ x x x + zα / 2 ⋅ σ x • 95% Confidence Interval: x − 1.96 ⋅ (σ x ) x x + 1.96 ⋅ (σ x ) E = zα / 2 ⋅ σ x = zα / 2 ⋅ σX n Margin of Error. 1 The Sampling Distribution for Text costs (n=16): Sampling (n=16): We determined previously that 80% of the sample means will fall between the values shown below. $225.89 $320.11 162.75 199.5 236.25 273 309.75 346.5 383.25 The Key to Confidence Intervals The The methods we use to develop the confidence interval give us random intervals: – Draw random sample – Estimate 3. Margin of Error and Sample Size. Margin of Error: E = zα / 2 ⋅ σx n = zα / 2 ⋅ σ x Illustrate E: 2 Reducing Margin of Error – what options? E = zα / 2 σx n Determining Sample Size • Given a desired level of confidence (eg. 90%) • Willing to accept a certain margin of error. • What sample size do you need? What is n? E = zα / 2 σx n σx is not usually known • Estimate – draw a small pilot sample just to estimate σx • Use the range (R) 3 4. Confidence interval - σx is not known. • We don’t know µ and we don’t know σ. 4. Confidence interval - σx is not known. t-distribution: a lot like the Z-distribution, but… • The t-distribution - “fatter tails” than the z. • More conservative – wider intervals for (1–α) • Reflects added uncertainty - also estimated σx . • Shape depends on degrees of freedom: df = (n – 1); • Center is zero • As (n – 1) → ∞ , the t-distribution converges to a z-distribution. 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 a 90% confidence interval estimate for the population mean. 4 ...
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