Bootstrap Method Introduction The bootstrap developed by Efron

Bootstrap Method - Introduction • The bootstrap, developed by Efron in the late 1970 s, allows us to calculate estimates in situations where there is no adequate statistical theory. • Example: calculating a CI for the mean when the population is not normal and the sample size is small. • Example: Calculating CI for other parameters such as the population median or other percentiles. week 7 1

Bootstrap Sample • Suppose we obtained a sample of n observations from some unknown distribution, F. We call the original sample X. • Our goal is to know about a parameter, θ, of the original distribution (for example: mean, median, standard deviation, upper quartile etc. ) • A bootstrap sample is a sample with replacement of size n from the original sample. It is denoted by X*. week 7 2

The Bootstrap Method • Use Minitab to sample B bootstrap samples of X, X*1, X*2, …, X*B. • Use Minitab to calculate an estimate of the parameter of interest θ from each of the B bootstrap samples. Call these estimates • The collection of estimates of the distribution of. form the bootstrap estimate • Use Minitab to calculate summary statistics and histograms for the bootstrap estimate of the distribution of. • Determine the bootstrap percentile interval for θ by finding the lower αB/2 percentile and the upper αB/2 percentile of the bootstrap distribution. week 7 3

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Hypotheses Testing • A hypothesis test is a formal procedure for comparing observed data with a hypothesis whose truth we want to assess. The hypothesis is a statement about the parameters in a population or model. • There are two types of hypotheses: Ø The null hypothesis, H 0, is the current belief. Ø The alternative hypothesis, Ha, is your belief, it is the negation of the null hypothesis. • The test of significance is designed to assess the strength of the evidence against the null hypothesis. week 7 5

Example Each of the following situations requires a significance test about a population mean . State the appropriate null hypothesis H 0 and alternative hypothesis Ha in each case. (a) The mean area of the several thousand apartments in a new development is advertised to be 1250 square feet. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicion. Answer: H 0: = 1250 ft 2 ; Ha: < 1250 ft 2 (b) Larry's car consume on average 32 miles per gallon on the highway. He now switches to a new motor oil that is advertised as increasing gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if his gas mileage actually has increased. Answer: H 0: = 32 mpg; Ha: > 32 mpg (c) The diameter of a spindle in a small motor is supposed to be 5 millimeters. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target. Answer: H 0: = 5 mm; Ha: 5 mm. week 7 6

Hypothesis Test Procedure • A hypothesis test is a proof by contradiction: we assume the null hypothesis is true, but we want to find evidence that does not support this assumption. • A hypothesis test is conducted by constructing a test statistic that incorporate both H 0 and the sample data and has a known distribution (usually Z, t, F, χ2). • The hypothesis test conclusion is based on how likely the observed test statistic (or more extreme) is under the assumption that H 0 is true. week 7 7

Rejection Region approach for Test Conclusion • There are two ways to determine if the test statistic is likely or not. • The first method is using a significant level, α, and a rejection region. If the test statistic is in the rejection region we have evidence against H 0. If the test statistic is not in the rejection region we find have no evidence against H 0. week 7 8

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