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Chapter 10 Parameter Estimation

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Alternatives to Hypothesis Testing? Some people say that the analysis I just presented, as well as some other things, is ample enough evidence to reject (ha ha) null hypothesis testing altogether, and opt for Parameter estimation

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Point estimation The question for point/parameter estimation, is: can we guess μ? Note that with hypothesis testing, we conclude (hopefully) that the probability of our results is low given that the null is true But, we can always draw the incorrect conclusion With point/parameter estimation, we specify value(s) which are likely to include μ

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Point estimation What’s the best single estimate of μ X (the population mean of the distribution from which we pulled our sample)? The sample mean is known as the “unbiased estimate of the population mean” “unbiased” because it neither systematically underestimates nor overestimates the population mean

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Biased Estimates The best estimate for population variance, is sample variance:

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Biased Estimates The formula for population variance is: But, when used to calculate sample variance, it overestimates the value, systematically, by a factor of 1 in the denominator, hence the term n-1

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Point estimation How good an estimate of μ is M X ? It’s the best, man. Yeah, but, it probably isn’t μ. So, what’s your point?

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Parameter Estimation Specifies a range of values, we call the “confidence interval” Then we can express with some degree of “confidence” that our interval contains μ Typically, people use 95% or 99% confidence intervals

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Method 1.State and Check Assumptions 2.Set Confidence Level 3.Obtain Independent Random Sample 4.Construct Confidence Interval 5.Interpret the Interval

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An Illustration One troubling aspect of IQ tests is that mean IQ scores increase with time. That is, when a new IQ test comes out, scores on the test systematically increase over the years until they are “re- standardized.” After this standardization, mean IQ scores increase again. –Cultural bias? –Teaching to the test? –Tester expectations?

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A school psychologist… Doesn’t trust the standards specified in the testing materials (because it is an old test), namely that μ = 100 She thinks that the standard deviation is probably unchanged, however (σ = 16) In this case, then, a hypothesis test that includes μ > 100 might be meaningless because we would expect the scores to increase with time anyways

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Can we estimate μ? The school psychologist opts to estimate μ instead With this information, she can compare schools, grade levels, etc., because she has a good estimate of μ She selects 25 4 th graders randomly from her district and gives them an IQ test,

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Parameter Estimation 1.State and check assumptions –Assumptions about the population Normally distributed – yes Variance - μ is known (16) –Assumption about the data Interval/Ratio level - OK –Assumptions about the sample Independent random sample – got it

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Parameter Estimation 2.Set a confidence level α =.05 1 – α =.95

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Parameter Estimation 3.Independent random sample –Got it

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Parameter Estimation 4.Construct confidence interval

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Parameter Estimation

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Thus, the 95% confidence interval is 99.728 ≤ μ ≤ 112.272

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Parameter Estimation 5. Interpret the confidence interval The probability is.95 that the interval 99.728 – 112.272 includes μ Our interpretation, though, is about the interval, not μ It would be incorrect to say that we have a 95% chance that μ is within the interval 99.728 – 112.272 μ doesn’t change, but M X does, as does the confidence interval from sample to sample

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10.1 – Estimating with Confidence. Recall: The Law of Large Numbers says the sample mean from a large SRS will be close to the unknown population mean.

10.1 – Estimating with Confidence. Recall: The Law of Large Numbers says the sample mean from a large SRS will be close to the unknown population mean.

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