IND212_Statistics_5th_Week - IND 212 Statistics Durdu Hakan...

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IND 212 Statistics Durdu Hakan Utku, Ph.D. 5th Week Spring 2016-2017
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Single Sample: Estimating the Mean One sided confidence bounds One-sided confidence bounds are developed in the same fashion as two-sided intervals One-sided probability statement that makes use of the Central Limit Theorem ܲ ିఓ < ܼ = 1 − ߙ By manipulating the equation P ߤ > ܺ ܼ = 1 − ߙ
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Single Sample: Estimating the Mean One sided confidence bounds Similar manipulation can be done for the equality ܲ ܺ − ߤ ߜ ݊ < −ܼ = 1 − ߙ And we get P ߤ < ܺ + ܼ = 1 − ߙ
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Single Sample: Estimating the Mean One sided confidence bounds on ࣆ, ࣌ known If ܺ is the mean of a random sample of size n from a population with variance ߪ , the one-sided 100(1 − α )% confidence bounds for μ are given by Upper one sided ݔ̅ + ݖ Lower one sided ݔ̅ − ݖ
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Single Sample: Estimating the Mean Example 4 In a psychological testing experiment, 25 subjects are selected randomly and their reaction time, in seconds , to a particular stimulus is measured. Past experience suggests that the variance in reaction times to these types of stimuli is 4 ࢙ࢋࢉ and that the distribution of reaction times is approximately normal . The average time for the subjects is 6.2 seconds . Give an upper 95% bound for the mean reaction time.
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Single Sample: Estimating the Mean Sol’n
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Single Sample: Estimating the Mean Sol’n The upper %95 bound: ݔ̅ + ݖ = 6.2 + 1.645 ଶହ = 6.2 + 0.628 = 6.858 That means that we are 95% confident that the mean reaction time is less than 6.858 seconds.
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Single Sample: Estimating the Mean The Case of σ Unknown If we have a random sample from a normal distribution , then the random variable ܶ = ିఓ has a Student t-distribution with n − 1 degrees of freedom. Here S is the sample standard deviation.
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Single Sample: Estimating the Mean The procedure is the same as that with σ known except that σ is replaced by S and the standard normal distribution is replaced by the t -distribution
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