Chapter 8 - Introduction to Statistical Inferences Chapter...

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Chapter 8 Introduction to Statistical Inferences
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Topics 1) Inference: Point Estimates and Interval Estimates 2) Confidence Intervals 3) Hypotheses 4) 1 st statistical test of μ
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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
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Statistics vs. Parameters Sample Statistics sample mean sample standard deviation sample variance Population Parameters population mean population standard deviation population variance INFERENCE Review
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How good is the point estimate? You never really know! Try to make sure point estimate is: how ?
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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
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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.
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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
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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
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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%
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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
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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)
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So now we can come up with interval that contains some % of all possible sample means x
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Calculating Confidence Intervals if σ is known (assumed in Chapter 8), we calculate the CI as: Use table 4 0 z 2 / 1 α = tail in area z (0.025) -(z (0.025) )
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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 each side is 5% 0 z 1 - α α / 2 α / 2 - (z (α/229 29 z (α/229 Table 4 with the z values already calculated
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Table 4
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