sLecture5dPBS - Lecture #5 Learning Objectives 1. Learn...

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Unformatted text preview: Lecture #5 Learning Objectives 1. Learn about the t-distribution and the role it plays in inference about a population mean when the population standard deviation is unknown. 2. Be able to calculate a confidence interval to estimate the mean when σ is not known. 3. Be able to perform a hypothesis test about the population mean when σ is not known. 4. Understand the matched (paired) samples procedures for comparing two groups with respect to the mean. 5. Know the definition of the following terms: Sample Standard Deviation Degrees of Freedom Student’s t Matched (paired) Samples “A single death is a tragedy; a million deaths is a statistic.”- Joseph Stalin (1879 - 1953) BM330 – Lecture 5 Inference for μ When X is Normal and σ is Unknown In most “real” inference problems, if the population mean is unknown, the population standard deviation is also unknown. What adjustments are required if the sample standard deviation is used as an estimate of the population standard deviation, i.e., we substitute s for σ ? Recall that our methods thus far have exploited the standard Normal probability distribution: Z = n X σ μ − ~ N(0, 1) The question, then, is what distribution results when we standardize X using an estimate of its standard error? As long as X has a Normal distribution (the underlying population is Normal or n is large), the result of the above standardization is a Student’s t random variable. n S X μ − = t Note that t measures “# of standard deviations” just as Z does Note that this result is exact if the underlying population is Normal. If not, this result will be approximate for “large” n . The t-distribution has only one parameter called the degrees of freedom . In one sample case, it is equal to ____. 1 BM330 – Lecture 5 The t-distribution is shown below in comparison to the Standard Normal: Z 0 Standard Normal t ( df = 5) t ( df = 13) Observations regarding the features of the t-distribution: Shape: Range of values for t: E[t] = μ t : Variability: In appearance, the t-distribution differs from the Z-distribution only in dispersion. The t-distribution has heavier tails. This makes sense, since we have introduced an additional source of variation, S , into the process. 2 BM330 – Lecture 5 Key Observations: 1. Shape 2. Number of standard deviations 4.04 4.02 4.00 3.98 3.96 0.95 0.025 0.025 X _______________ ׀ ______________ m m ׀ ________________ t - t α / 2 t α /2 Result: The ideas that we have used to develop our confidence interval estimate and hypothesis testing “formulas” remain the same. We need only substitute S for σ and a “t-score” for the “Z-score.” Some example t-critical scores for various cases area given below: Sample Size Confidence Level t- Score Z- Score 10 95% 2.262 1.96 21 99% _____ 2.576 17 90% 1.746 1.645 16 90% _____ 1.645 3 BM330 – Lecture 5 (1 – α )% Confidence Interval estimates of μ when σ is unknown and X has a Normal distribution: ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛...
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This note was uploaded on 02/22/2009 for the course BUSMGT 330m taught by Professor Kriska during the Spring '08 term at Ohio State.

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sLecture5dPBS - Lecture #5 Learning Objectives 1. Learn...

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