Testing Hypotheses I Lesson 9 Descriptive vs. Inferential Statistics Descriptive quantitative descriptions of characteristics Inferential Statistics Drawing conclusions about parameters ~ Hypothesis Testing Hypothesis testable assumption about a parameter should conclusion be accepted? final result a decision: YES or NO qualitative not quantitative General form of test statistic ~ Hypothesis Test: General Form systematicvariation test statistic unsystematic variation difference between groups effect test statistic difference due to chance error zobs X X Evaluating Hypotheses Hypothesis: sample comes from this population Two Hypotheses Testable predictions Alternative Hypothesis: H1 also scientific or experimental hypothesis there is a difference between groups Or there is an effect Reflects researcher’s prediction Null Hypothesis: H0 there is no difference between groups Or there is no effect This is hypothesis we test ~ Conclusions about Hypotheses Cannot definitively “prove” or “disprove” Logic of science built on “disproving” easier than “proving” State 2 mutually exclusive & exhaustive hypotheses if one is true, other cannot be true Testing H0 Assuming H0 is true, what is probability we would obtain these data? ~ Hypothesis Test: Outcomes Reject Ho accept H1 as true supported statistical significance by data difference greater than chance Fail to reject “Accepting” Ho data are inconclusive ~ Hypotheses & Directionality Directionality affects decision criterion Direction of change of DV Nondirectional hypothesis Does reading to young children affect IQ scores? Directional hypothesis Does reading to young children increase IQ scores? ~ Nondirectional Hypotheses 2-tailed test Similar to confidence interval Stated in terms of parameter Hypotheses H1 : 100 Ho : = 100 Do not know what effect will be can reject H0 if increase or decrease in IQ scores ~ Directional Hypotheses 1- tailed test predict that effect will be increase or decrease Only predict one direction Prediction of direction reflected in H1 H1: > 100 Ho: < 100 Can only reject H0 if change is in same direction H1 predicts ~ Errors “Accept” or reject Ho only probability we made correct decision also probability made wrong decision Type I error (a) incorrectly rejecting Ho e.g., may think a new antidepressant is effective, when it is NOT ~ Errors Type II error (b) incorrectly “accepting” Ho e.g., may think a new antidepressant is not effective, when it really is Do not know if we make error Don’t know true population parameters *ALWAYS some probability we are wrong P(killed by lightning) 1/1,000,000 p = .000001 P(win powerball jackpot) 1/100,000,000 ~ Errors Actual state of nature H0 is true Reject H0 H0 is false Type I Error Correct Correct Type II Error Decision Accept H0 Definitions & Symbols a Level of significance Probability of Type I error 1-a Level of confidence b Probability of Type II error 1-b Power ~ Steps in Hypothesis Test 1. State null & alternative hypotheses 2. Set criterion for rejecting H0 3. Collect sample; compute sample statistic & test statistic 4. Interpret results is outcome statistically significant? ~ Example: Nondirectional Test Experimental question: Does reading to young children affect IQ scores? = 100, = 15, n = 25 We will use z test Same as computing z scores for X ~ Step 1: State Hypotheses H0: = 100 Reading to young children will not affect IQ scores. H1: 100 Reading to young children will affect IQ scores. ~ 2. Set Criterion for Rejecting H0 Determine critical value of test statistic defines critical region(s) Critical region also area of distribution beyond critical value in called rejection region tails If test statistic falls in critical region Reject H0 ~ 2. Set Criterion for Rejecting H0 Level of Significance (a) Specifies critical region area in tail(s) Defines low probability sample means Most common: a = .05 others: .01, .001 Critical value of z use z table for a level ~ Critical Regions a = .05 zCV = + 1.96 f -2 -1.96 -1 0 +1 +2 +1.96 3. Collect data & compute statistics Compute sample statistic X Observed value of test statistic zobs X X Need to calculate X ~ 3. Collect sample & compute statistics 100, 15 assume : X 105.5 X zobs n X X n = 25 15 3 25 105 .5 100 5.5 1.83 3 3 Critical Regions a = .05 zCV = + 1.96 f -2 -1.96 -1 0 +1 +2 +1.96 4. Interpret Results Is zobs in the critical region? NO we fail to reject H0 These data suggest reading to young children does not affect IQ. No “significant” difference does not mean they are equal data inconclusive ~