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Why do we conduct hypothesis tests anyway? 1.  Helps us generalize beyond our data •  …but before we do this, we must always measure the effect of chance, that is, we must obtain a value for standard error 2.  Helps us minimize the chances of drawing an incorrect conclusion More Hypothesis Testing! Double the fun Do you remember our crime example? What if we had used α = .01? ▪  Our sample mean and our z-­‐value would still be the same, but the critical values of z that separate the region would be different, ±2.58 ▪  This is a stricter or more conservative value (it is harder to reject the null hypothesis) ▪  With this alpha level, you would fail to reject the null hypothesis Two-­‐tailed vs. One-­‐tailed test ▪  Two-­‐tailed: rejection ▪  One-­‐tailed: rejection ▪  H1: X ≠μ ▪  H1: X ≥μ OR X < μ regions are in both tails of the sampling distribution region is in just one tail of the sampling distribution Critical z-­‐values for one-­‐ and tw0-­‐tailed tests Level of Significance (α) .05 Two-­‐tailed (no direction) ±1.96 One-­‐tailed (lower critical tail) -­‐1.65 One-­‐tailed (upper critical tail) +1.65 .01 ±2.58 -­‐2.33 +2.33 Example! ▪  You want to know whether people with cats have less stress than the general population. You poll 36 cat owners and they have an average stress level of 20. The average stress of the population is 30 with a standard deviation of 10. 3. Calculate z-­‐value 1. Formulate hypotheses: ▪  H0: cat owners are just as stressed or more stressed than everyone else 20 − 30 −10
Z=
=
= −6.0
10
1.667
36
X >µ
▪  H1: cat owners are less stressed than everyone else X <µ
2.  Decision rule: •  α = .05 •  Critical Z = ±1.65 4.  Make a decision and interpret Z observed < Z critical -­‐6 < -­‐1.65 Reject the null! Cat owners report less stress! Practice! ▪  One-­‐tailed test, lower critical tail, α = .01 ▪  z = -­‐1.97 ▪  z = -­‐5.00 ▪  z = -­‐3.01 ▪  Two-­‐tailed test, α = .01 ▪  z = -­‐2.33 ▪  z = 3.00 ▪  z = -­‐1.97 Level of Significance (α) .05 .01 Two-­‐tailed (no direction) ±1.96 ±2.58 One-­‐tailed (lower critical tail) -­‐1.65 -­‐2.33 One-­‐tailed (upper critical tail) +1.65 +2.33 ▪  One-­‐tailed test, upper tail critical, α = .05 ▪  z = -­‐1.61 ▪  z = -­‐1.88 ▪  z = 1.96 Sorry…z tests are not always the answer ▪  In a z-­‐test, you compare your sample to a known population with a known mean and standard deviation ▪  In real research practice, you often compare two or more groups of scores to each other without any direct information about populations ▪  Nothing is known about populations that the samples are supposed to come from… Four possible outcomes of a hypothesis test Reality Decision Retain H0 H0 true H0 false Correct Decision Miss: Type II Error 1-­‐α β Reject H0 False Alarm: Type I Error α Correct Decision 1 – β (power) ▪  Type I Error: rejecting a true null hypothesis ▪  Alpha (α): the probability of a Type I error ▪  Type II Error: retaining a false null hypothesis ▪  Beta (β): The probability of a Type II error Online Resources: http://www.youtube.com/watch?
v=plmu-­‐64iq84 & http://www.youtube.com/watch?
v=iz1sfne1cNA&feature=related Example 1 Example 2 Example 3 Summary: The story of Hypothesis Testing …sometimes called Null Hypothesis Significance Testing (NHST) ▪  Start by assuming the null hypothesis is true (H0) ▪  i.e., there is no relationship between the variables of interest in a population ▪  Collect data from a sample that represents the population of interest ▪  Calculate an observed test statistic (e.g., z, t, F) ▪  Determine the probability that this observed test-­‐statistic is “true” to determine if the null hypothesis is true ▪  A large summary statistic means there is a small probability of H0 being true!!! ▪  i.e., the bigger the absolute value of your test-­‐statistic is, the less likely that the null hypothesis is true Type I errors are all around you! When you have to accept the null hypothesis… Fashion Recommendations