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. Review: Sampling Distribution • Standard Error: the standard deviation of the sampling distribution σx = σx n If the sampling distribution is normal. 1 Suppose you draw a sample (n = 9) from a normal (n 9) population with μ = 10 and σ = 3. Before drawing your Before sample, the probability of getting a sample that gives you a sample mean between 8.04 and 11.96 miles is… (Enter is… the probability – use 2 decimals.) probability P(8.04 < X < 11.96) ResEc 211 student textcosts (X) for Fall 2009 are normally $273 distributed with μX = $273 and σX = $147. Before drawing a sample of size 16, P(a < < b) = 0.80. X Determine the lower limit a for this symmetric interval. Round Round to a whole number. 162.75 199.5 236.25 273 309.75 346.5 383.25 PRS 5: Enter your sample number from today’s sample from handout. This sample number will be used to match your responses to the correct values in order to grade your PRS questions (Duplicate order to grade your PRS questions. (Duplicate sample sample numbers will result in grades of “0.”) 2 PRS 6: Use your sample of 16 text costs to estimate the population mean textbook costs. Report your estimate to 2 decimal places ($$$.¢¢). PRS PRS 7: Did your sample mean fall within the 0.80 probability interval we constructed? Enter “0” for No; Enter “1” for Yes. 162.75 199.5 236.25 273 309.75 346.5 383.25 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?” 3 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 on your Center that interval on your 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. Confidence Intervals – σX known • 95% Confidence Interval: 4 PRS 7: 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.) U- 162.75 199.5 236.25 273 309.75 346.5 383.25 PRS 8: Does your 80% confidence interval contain the true value, µ = $273? If No, enter “0” “0” If Yes, enter “1” 5 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. 6 ...
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