STA309 - Final cheat sheet

# STA309 - Final cheat sheet - STA309 Final Inference for...

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STA309 Final Inference for Distributions Inference for the mean of a population: o When σ is known: Tests and confidence intervals for the mean m of a normal population are based on the sample mean J of an SRS. Because of the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample is large. o The standardized sample mean is the one-sample z statistic (When we know s we use the z statistic and the standard normal distribution.): Assumptions: SRS; n < 15, normal population; n ≥ 15; no outliers or strong skewness estimate = J SE J = σ / √n critical value = z* o When σ is unknown: In practice, we do not know s. Assumptions: SRS; n < 15, data close to normal; 15 ≤ n <40, no outliers or strong skewness; n ≥ 40; even for clearly skewed data estimate = J SE = s / √n. Get the one-sample t statistic (with n -1 degrees of freedom): Critical value = t*(n - 1) The t distribution : o There is a t distribution for every positive integer k (degree of freedom). Call the distribution t(k). o The density curves of the t(k) distribution are symmetric about zero and bell-shaped. o The curves are more spread out than the standard normal curves (due to substituting s for s). o *** As the degrees of freedom k increase, the t(k) density curve approaches the N(0,1) curve o One-sample t procedures : A level C confidence interval for μ (look at Table D for t* using df = n – 1): Hypothesis testing of H 0 : μ = μ 0 . 1. State the hypothesis. 2. Calculate the test statistic (solve for SE, find t* from Table D and α, plug into CI equation). 3. Compute the P -value. Remember to * 2 if two tailed test! 4. Compare the P -value to α and make your conclusion. (p-value ≤ α reject H 0 ; p-value > α accept H 0 ) Robustness of t procedures : a statistical inference procedure is robust if the probability calculations required are insensitive to violations of the conditions that usually justify the procedure. The t procedures are relatively robust when the population is non-normal, especially for larger sample sizes. The t procedures are useful for non-normal data when n ≥ 15 unless the data show outliers or strong skewness. n large and equal Two Population Means: o Matched Pairs Comparisons: subjects are matched in pairs and each treatment is given to one subject in each pair. W e have a separate sample from each treatment or each population. Subjects are matched in pairs and each treatment is given to one subject in each pair. To compare the responses to two treatments, apply the one-sample t procedures to the observed differences . We calculate the difference in “before” and “after” test scores for each in the pair. Assumptions:

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STA309 - Final cheat sheet - STA309 Final Inference for...

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