Chapter 8

# 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
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
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
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
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%
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)
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) )
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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