Hypothesis Testing - Study Guide Part 1

Hypothesis Testing ______________________________________________ 1) Compare the statistical questions that hypothesis tests are designed to address with those that confidence intervals are designed to address. 2) Define the Null and Alternative Hypotheses 3) Explain why we always start by assuming the Null Hypothesis is true 4) Review the basic steps for conducting a hypothesis test 5) Present terminology for describing the results of hypothesis tests 6) Discuss the kinds of errors that hypothesis test are liable to produce 7) Understanding how α functions in the context of hypothesis testing. 8) Outline the basic steps for conducting a hypothesis test. Saving the environment ______________________________________________ Jake is the manager of ε, the cafeteria at Stat State University. He is environmentally conscious so he is always looking for ways to conserve resources. He thinks that placing napkin dispensers on every table rather than at the silverware station might reduce the number of napkins that people used. To test his hypothesis, Jake could run an experiment in which he moved the napkin dispensers and measured napkin consumption for a period of time. Q: What would he do with his sample data? A: He could calculate a confidence interval, but  Jake doesn’t care about µ  Jake cares about whether moving the napkin dispenser affected napkin consumption Q: So, what can Jake do? A: Hypothesis Testing What is Hypothesis testing? ______________________________________________ Does the sun revolve around the earth? The church said ‘YES’, but Copernicus wasn’t convinced, so he collected and analyzed tons of data and compared those data against two competing hypotheses: a) The sun revolves around the earth b) The earth revolves around the sun This is the basic approach of hypothesis testing Key Terms ______________________________________________ Hypothesis Testing: a statistical method for deciding which of two hypothetical outcomes for an experiment is more consistent with the experimental data. Statistical Hypothesis: an educated guess about the value of a population parameter on the basis of past experience or data collection Null Hypothesis: a statistical hypothesis that suggests that an experimental manipulation will not affect the outcome of an experiment. The null hypothesis is usually denoted with either HO or H0. Alternative Hypothesis: a statistical hypothesis that suggests that an experimental manipulation will affect the outcome of an experiment. The alternative hypothesis is usually denoted with either HA or H1. The Null and Alternative Hypotheses ______________________________________________ Label Common Language Statistical Language Statistical Notation Moving the napkin Null dispensers Hypothesis will not affect consumption. µ will remain at 100 pounds after the napkin dispensers are moved HO: µ = 100 Moving the napkin Alternative dispensers Hypothesis will affect consumption. µ will no longer be 100 pounds after the napkin dispensers are moved HA: µ ≠ 100 Assuming the Null Hypothesis is True ______________________________________________ What is the guiding principal of our legal system?  Innocent until proven guilty beyond the shadow of a doubt What is the guiding principal of hypothesis testing?  The null hypothesis is assumed to be true unless there is overwhelming evidence to the contrary. Professor Hobbes suspects that two students cheated on his philosophy exam: Null Hypothesis (Ho): The students did not cheat Alternative Hypothesis (Ha): The students did cheat Evidence:  Same grade  Missed same questions  Gave same incorrect answers to all questions they missed The Logic of Hypothesis Testing ______________________________________________ Jake’s napkin dispensers Ho: µ = 100 Ha: µ ≠ 100 Collect a sample of data and compare it with the two hypotheses. What would you conclude if his sample mean was:  99.8 pounds?  97 pounds?  95 pounds?  90 pounds? At some point, the sample mean would be so far from 100, that we could not believe the null hypothesis was true. In other words, at some point, we would have to conclude that our sample mean is such a rare event that we must abandon the null hypothesis.  How rare does an event have to be? o It’s up to you…sort of. o The probability of observing our sample mean has to be less than α. Why α? ______________________________________________ What is α? It is a measure of reliability, just as it was with confidence intervals. What does ‘measure of reliability’ mean? It tells us the probability that the statistical inference we make will be an error. If we set α = .05:  We are 95% sure that our confidence interval will contain µ.  We are 95% sure that the decision we make about the null will be correct. In both cases, there is a 5% chance that we will make an error. Statistical inferences are never deterministic. We can never be 100% sure of any statistical inference. The basic steps for conducting Hypothesis Tests ______________________________________________ 1) Use α to determine the critical value of z (zcrit). a. zcrit is often called a cutoff score. 2) Use the sample data to calculate the observed value of z (zobs). 3) Compare zobs with zcrit: a. If |zobs| > |zcrit|, we conclude that our sample mean is a rare event b. If |zobs| ≤ |zcrit|, we conclude that our sample mean is a not rare event The basic steps for conducting t-tests 1. Determine the value for zcrit 2. Calculate zobs 3. Compare zobs with zcrit Basic Steps of Hypothesis Tests: Moving the Napkin Dispensers ______________________________________________ Jake moved the napkin dispensers in ε to determine if it would reduce napkin consumption. Based on past experience, Jake knows that ε diners use 100 pounds of napkins per day; this means that µ0 = 100. Jake also knows from past experience that σ = 20 pounds. After moving the napkin dispensers, Jake collected data on napkin use over the next 100 days and finds a sample mean of 95.5 pounds. Do these data suggest that moving the napkin dispensers affected consumption if we set α = .05? Step 1: Determining a value for zcrit ______________________________________________ Find the value for z that leave α/2 in each tail. If α = .05, we find the z-score that leaves .025 in each tail. Step 1: Determining a value for zcrit ______________________________________________ Find the value for z that leave α/2 in each tail. If α = .01, we find the z-score that leaves .005 in each tail. Step 2: Calculating a value for zobs ______________________________________________ Z-score Formula (Chapter 4) z z obs  Zobs Formula M  / n M  0 / n (µ0: the value of µ according to the null hypothesis) z obs  M  0 / n z obs  95.5  100 z obs   4.5 z obs   4.5 20 / 100 20 / 10 2 zobs = -2.25 (Remember: the sign matters when calculating z-scores!)