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Stat 344 Lecture 20 (20, 21)

# Stat 344 Lecture 20 (20, 21) - Confidence Intervals Lecture...

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1 Confidence Intervals Lecture 20 Topics Variances Proportions Prediction intervals Lecture 19 R eference: Montgomery Sections 9-1 thru 9-3 Devore Lecture 20 Devore Lecture 21

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Hypothesis and Test Procedures Lecture 20 Topics Hypothesis tests versus confidence intervals. How to write hypothesis statements Type I and Type II errors & their probabilities. Concept of the critical cutoff statistic & the rejection region. Defined alternative hypotheses Lecture 20 Reference: Devore Sec 8.1 Hypothesis and test procedures 2 Stat 344 Lecture 20
What are Hypotheses Hypothesis: A claim or assertion about a population parameter. H 0: is the symbol for the null hypothesis, The current value, The status quo. H a: is the symbol for the alternative hypothesis, aka H 1:, regards a claim. Stat 344 Lecture 20 3

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Hypothesis Structure H 0: μ = 90, 90 is the hypothesized value of μ. Parameter could be σ2 or p. H a: μ ≠ 90 < for a left-tail test > for a right-tail test ≠ for a two-tail test We seek to reject the null hypothesis if there is sufficient evidence to do so. Stat 344 Lecture 20 4
Hypothesis Decision Matrix α = Prob(Type I error). Type I error is the error most to be avoided. β = Prob(Type II error). Stat 344 Lecture 20 5 Our decision H0 is true μ = 90 Ha is true μ ≠ 90 Do not reject H 0 OK Type II error Reject H 0 Type I error OK Reality

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Example 8.1 Bumper Damage-a An car bumper sustains no visible damage 25% of the time in a crash test. A proposed redesign seeks to increase that percentage. The redesign will be tested with a sample of n =20 crashes. H 0: p = 0.25, H a: p > 0.25 Stat 344 Lecture 20 6 X is the number of crashes with no visible damage, a binomial rv. Rejection region R 8 = {8, 9, 10,…,19, 20} when x ≥ 8 and H 0: p = 0.25. α = P (Type I error) = P ( H 0 rejected when true) = P ( x ≥ 8) = 1– P ( x ≤ 7) = 0.102 x P(X ≤ x) 0 0.003 1 0.024 2 0.091 3 0.225 4 0.415 5 0.617 6 0.786 7 0.898 8 0.959 9 0.986 10 0.996 20 1.000 B(n=20, p=0.25 0.20 0.15 0.10 0.05 0.00 X Probability 8 0.102 0 Binomial, n= 20, p= 0.25
Example 8.1 Bumper Damage-b Now suppose H a: p =0.3 β(0.3) = P (Type II error when p = 0.3) = P ( H 0 is not rejected when it is false because p = 0.3) = P(x ≤ 7) = B(7, 20, 0.3) = 0.772 Stat 344 Lecture 20 7 0.20 0.15 0.10 0.05 0.00 X Probability 7 0.132 17 Binomial, n= 20, p= 0.5 p β(p) 0.3 0.772 0.4 0.416 0.5 0.132 0.6 0.021 0.7 0.001 0.8 0.000

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Example 8.2 Paint Drying Time-a The drying time of a paint is known to be a normally distributed rv with μ = 75 min and σ = 9 min . An additive is proposed to decrease the drying time. Because of the high cost of the additive, there should be strong evidence that it decreases the drying time. Test will be based on 25 samples of paint with the additive.
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