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Unformatted text preview: Chapter 8 Introduction to Statistical Inferences Topics • 1) Inference: Point Estimates and Interval Estimates • 2) Confidence Intervals • 3) Hypotheses • 4) 1 st statistical test of μ Inference • Remember the steps: – Identify population of interest – Take a sample – Use sample data to make ‘best guess’ of population value • Sample value is a Point Estimate for a Parameter Sample Data used to Estimate ►True Population Values Statistics vs. Parameters • Sample Statistics • sample mean • sample standard deviation • sample variance • Population Parameters • population mean • population standard deviation • population variance INFERENCE Review How good is the point estimate? • You never really know! • Try to make sure point estimate is: – • how ? Point estimates are just that  points • You can support the point estimate with more information • Interval Estimate – a range of values that you have ‘some confidence’ includes the true population value • Confidence Intervals for true population mean 2) Confidence Intervals Conceptually • We have talked about generating the Sampling Distribution of Sample Means (SDSM) • The distribution of all possible means that could be sampled from a population • The SDSM will be normally distributed, even if the population isn’t, when n > 30. • If the SDSM is normally distributed, 95% of the possible sample means will fall within approximately 2 standard errors of the mean. Sampling distribution of sample means x μ μ 2 x σ μ + 2 x σ x α • We will define the area under the curve in the tails as equal to a value of α • So when there are 2 tails, ½ of area in each tail α /2 α /2 So if the middle has 95% of the values…. • then there is 0.05/2 = 0.025 or 2.5% in each tail, so total alpha is 5% So we can use this information to create a Confidence Interval (CI) • This is an interval (between 2 numbers) that we are ‘ some level of confidence ’ sure will include the true population value (mean) • calculated with sample information • Note: 100% CI will go from –infinity to positive infinity Confidence Interval for μ • This is the common type of Confidence Interval • can use z table (if population is normally distributed) or n greater than or equal to 30 • We will assume that our sample mean is a best guess of μ • For Chapter 8, we will assume that the POPULATION STANDARD DEVIATION IS KNOWN (Usually don’t know σ – but we will get to that in Chapter 9) So now we can come up with interval that contains some % of all possible sample means x Calculating Confidence Intervals • if σ is known (assumed in Chapter 8), • we calculate the CI as: • Use table 4 z 2 / 1 α = tail in area z (0.025)(z (0.025) ) 1 α Confidence Interval for μ • So if you want a 95% confidence interval then α = 5 % and the area under the curve on each side is 2.5% • If you want a 90% confidence interval then α = 10% and the area under the curve on...
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This document was uploaded on 11/04/2011 for the course BIOM 301 at Maryland.
 Fall '08
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