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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 μX
1. Point Estimate: a single numeric value – our best guess about μX • Draw a sample of data from population. x • Estimate: use sample data in estimator: x = ∑
n • • • Sampling Distribution – the sample mean is a Distribution the sample mean is CRV. Probabilities: probability a sample mean falls within a certain interval. σ If Sampling Distribution normal: σ x = x n We found 80.7% of our random sample means fell within the probability interval : P($ P($225.89 < X < $320.11) = 0.80? $320.11 Our theory would predict that 80% would fall in the interval.
$225.89 $320.11 162.75 199.5 236.25 273 309.75 346.5 383.25 1 2. Confidence Intervals – σX known.
• Suppose: 1. 1. We don’t know μX . (C’mon – pretend.) 2. We Do know σX
The The question you ask: ask: “How confident am I that my estimate is close enough to μX that my interval will fall over μX?” 2. Confidence Intervals – σX known
Sampling distribution: Probability interval for the sample mean: interval for the sample mean: P( μ x − zα /2 ⋅ σ x < x < μ x + zα / 2 ⋅ σ x ) = (1 − α ) Confidence Level – what do we mean by “confidence?”
Make Make an interval that is as wide as the 0.80 probability interval shown in our sampling distribution graph. Center that interval over your Center that interval over your point estimate. If If your mean falls in the green shaded area, your CI will “contain the population mean.” How How often will that happen? 80% of the time in repeated sampling. That’s That’s why we are 80% confident. 2 2. Confidence Intervals – σX known • 95% Confidence Interval: PRS 5: Calculate your 80% confidence interval your lower limit using your sample mean and our knowledge of the sampling distribution. Round Round to two decimal places to two decimal places. X ± zα / 2 ⋅ σ X Where do your mean and interval fall? (The interval is a wide as your U-card, the Ugreen/burgundy division on your U-card is Uthe center of your CI.) $225.89 $320.11 162.75 199.5 236.25 273 309.75 346.5 383.25 3 PRS 6: Does your 80% confidence interval contain the true value, µ = $273? If No, enter “0” If Yes, enter “1” 2. Confidence Intervals – σX known Confidence Interval – Interpretation The Key to Confidence Intervals Confidence
The The methods we use to develop the confidence interval give us random intervals: – Draw random sample – Estimate Proper interpretation: 95% of the samples we Proper interpretation: 95% of the samples we draw will result in a confidence interval that falls over (contains) the true population mean. Thus, we are 95% confident that our particular interval contains the true population mean. 4 3. Margin of Error and Sample Size. Margin of Error: E = zα / 2 ⋅ σx n = zα / 2 ⋅ σ x • Illustrate the Margin of Error: E = zα / 2 ⋅ σx
n = zα / 2 ⋅ σ x Reducing Margin of Error – what options? E = zα / 2 σx n 5 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) 6 ...
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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