Hypothesis Testing and Confidence Intervals in Statistics: MATH 243, Lecture 17, Study notes of Probability and Statistics

Download Hypothesis Testing and Confidence Intervals in Statistics: MATH 243, Lecture 17 and more Study notes Probability and Statistics in PDF only on Docsity! MATH 243, LECTURE 17 1. Testing hypotheses Our focus in these last weeks of class will be almost exclusively on confidence intervals and hypothesis testing. Last time we slowly developed the language and methods of basic hypothesis testing. We will review these now but be more direct. General setup: Suppose you are studying some random variable of some population. Suppose someone else makes a guess that the mean µ = µ0 for some number µ0. (And suppose that you know the standard deviation, σ for the population. We’ll remove this assumption when we study §16). You suspect that µ0 is incorrect; that is that µ 6= µ0. Your audience will be convinced, if you can show that P (µ = µ0) < α. (α is typically .05 or .1 or .01). Our setup is now: • H0 : µ = µ0. Null hypothesis. • Ha : µ 6= µ0. Alternative hypothesis. • Significance level α. To try to convince an audience that H0 is wrong (to reject the null hypothesis) we do the following: a. Take a SRS from population of size n. b. Calculate x (our sample mean) and the z-statistic z = x− µ0 σ/ √ n c. Calculate the “P-value:” = the probability (assuming H0 is true) that x is at least this far from µ0 = P (sample mean ≥ µ0 + |x− µ0|)+ P (sample mean ≤ µ0 − |x− µ0|) = P (Z ≥ |z|) + P (Z ≤ −|z|). d. If P ≤ α this is statistically significant at level α and leads you to reject the null hypothesis. If P > α this is not statistically significant at that significance level, which means that you can neither reject or accept the null hypothesis. NOTE: We are concentrating on the situation where H0 is of form µ = µ0. Ha can be serveral things. Either • Ha : µ 6= µ0 (if we don’t know which way µ might be off from µ0). This is called the two-sided alternative. • Ha : µ > µ0 (if we suspect µ might really be larger than µ0). This is called a one-sided alternative. • Ha : µ < µ0 (if we suspect µ might really be smaller than µ0). This is another one-sided alternative. 1