Example: Propellant Burn Rate

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Presentation transcript:

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.

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

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

Hypothesis Testing  = P(Type I error) = P(reject H0 | H0 is true)  = P(Type II error) = P(accept H0 | H0 is false) Decision H0 is true H0 is false Fail to reject H0 No error Type II error Reject H0 Type I error

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 ?

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 ?

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.