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Example: Propellant Burn Rate

Example: Propellant Burn Rate. Aircrew escape systems are powered by a solid propellant. Specifications require that the mean burn rate must be 50 cm/s. H 0 :  = 50 10 samples are tested. Definitions. Critical region : range of values for which the null hypothesis is rejected

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Example: Propellant Burn Rate

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  1. Example: Propellant Burn Rate • Aircrew escape systems are powered by a solid propellant. Specifications require that the mean burn rate must be 50 cm/s. • H0:  = 50 • 10 samples are tested.

  2. Definitions • Critical region: range of values for which the null hypothesis is rejected • Acceptance region: range of values for which the null hypothesis is not rejected • Critical values: boundaries between the critical and acceptance regions

  3. More Definitions • Type I error: rejecting the null hypothesis when it is true • Type II error: failing to reject the null hypothesis when it is false • Significance level: probability of type I error

  4. Hypothesis Testing •  = P(Type I error) = P(reject H0 | H0 is true) •  = P(Type II error) = P(accept H0 | H0 is false)

  5. Example Part (1) • Suppose the acceptance region is 48.5   51.5 • Suppose that the burn rate has a standard deviation of 2.5 cm/s, and has a distribution for which the CLT applies. • Find , P(Type I error) • What are some ways to reduce ?

  6. Example Part (2) • Suppose it is important to reject the null hypothesis when >52 or <48. • Let the alternate hypothesis, H1:  = 52 • Find , P(Type II error) • What affects the size of ?

  7. Interlude • Type I error can be directly controlled: rejecting the null hypothesis is a strong conclusion. • Type II error depends on sample size and the extent to which the null hypothesis is false: accepting the null hypothesis is a weak conclusion. • The power of a test is the probability of rejecting the null hypothesis when the alternative hypothesis is true, i.e., 1–. • The P-value is the smallest level of significance that would lead to rejection of the null hypothesis.

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