DIRECTIONAL HYPOTHESIS The 1-tailed test: –Instead of dividing alpha by 2, you are looking for unlikely outcomes on only 1 side of the distribution –No critical area on 1 side—the side depends upon the direction of the hypothesis –In this case, anything greater than the critical region is considered “non-significant”

Non-Directional & Directional Hypotheses Nondirectional –H o : there is no effect: (X = µ) –H 1 : there IS an effect: (X ≠ µ) –APPLY 2-TAILED TEST 2.5% chance of error in each tail Directional –H 1 : sample mean is larger than population mean (X > µ) –H o x ≤ µ –APPLY 1-TAILED TEST 5% chance of error in one tail

Why we typically use 2-tailed tests Often times, theory or logic does allow us to prediction direction – why not use 1- tailed tests? Those with low self-control should be more likely to engage in crime. Rehabilitation programs should reduce likelihood of future arrest. What happens if we find the reverse? –Theory is incorrect, or program has the unintended consequence of making matters worse.

STUDENT’S t DISTRIBUTION –We can’t use Z distribution with smaller samples (N<100) because of large standard errors –Instead, we use the t distribution: –Approximately normal beginning when sample size > 30 –Probabilities under the t distribution are different than from the Z distribution for small samples –They become more like Z as sample size (N) increases

THE 1-SAMPLE CASE –2 Applications Single sample means (large N’s) (Z statistic) –May substitute sample s for population standard deviation, but then subtract 1 from n »s/ √N-1 on bottom of z formula Smaller N distribution (t statistic), population SD unknown

STUDENT’S t DISTRIBUTION –Find the t (critical) values in App. B of Healey –“degrees of freedom” # of values in a distribution that are free to vary Here, df = N-1 –When finding t(critical) always use lower df associated with your N Practice: ALPHATESTNt(Critical).052-tailed tailed tailed tailed15

Example: Single sample means, smaller N and/or unknown pop. S.D. 1.A random sample of 26 sociology grads scored an average of 458 on the GRE sociology test, with a standard deviation of 20. Is this significantly higher than the national average (µ = 440)? 2.The same students studied an average of 19 hours a week (s=6.5). Is this significantly different from the overall average (µ = 15.5)? USE ALPHA =.05 for both

1-Sample Hypothesis Testing (Review of what has been covered so far) 1.If the null hypothesis is correct, the estimated sample statistic (i.e., sample mean) is going to be close to the population mean 2.When we “set the criteria for a decision”, we are deciding how far the sample statistic has to fall from the population mean for us to decide to reject H 0 –Deciding on probability of getting a given sample statistic if H 0 is true –3 common probabilities (alpha levels) used are.10,.05 &.01 These correspond to Z score critical values of 1.65, 1.96 & 258

1-Sample Hypothesis Testing (Review of what has been covered so far) 3.If test statistic we calculate is beyond the critical value (in the critical region) then we reject H 0 –Probability of getting test stat (if null is true) is small enough for us to reject the null –In other words: “There is a statistically significant difference between population & sample means.” 4.If test statistic we calculate does not fall in critical region, we fail to reject the H 0 –“There is NOT a statistically significant difference…”

2-Sample Hypothesis Testing (intro) –Apply when… You have a hypothesis that the means (or proportions) of a variable differ between 2 populations –Components –2 representative samples – Don’t get confused here (usually both come from same “sample”) –One interval/ratio dependent variable –Examples »Do male and female differ in their aggression (# aggressive acts in past week)? »Is there a difference between MN & WI in the proportion who eat cheese every day? –Null Hypothesis (H o ) The 2 pops. are not different in terms of the dependent variable

2-SAMPLE HYPOTHESIS TESTING Assumptions: –Random (probability) sampling –Groups are independent –Homogeneity of variance »the amount of variability in the D.V. is about equal in each of the 2 groups –The sampling distribution of the difference between means is normal in shape

2-SAMPLE HYPOTHESIS TESTING We rarely know population S.D.s –Therefore, for 2-sample t-testing, we must use 2 sample S.D.s, corrected for bias: »“Pooled Estimate” Focus on the t statistic: t (obtained) = (X – X) σ x-x we’re finding the difference between the two means… …and standardizing this difference with the pooled estimate

2-SAMPLE HYPOTHESIS TESTING t-test for the difference between 2 sample means: Addresses the question of whether the observed difference between the sample means reflects a real difference in the population means or is due to sampling error Sample Sampling Distribution – difference between sample means (closer sample means will have differences closer to 0) ASSUMING THE NULL!

Applying the 2-Sample t Formula –Example: Research Hypothesis (H 1 ): –Soc. majors at UMD drink more beers per month than non-soc. majors –Random sample of 205 students: »Soc majors: N = 100, mean=16, s=1.0 »Non soc. majors: N = 105, mean=15, s=0.9 »Alpha =.01 »FORMULA: t(obtained) = X 1 – X 2 pooled estimate

Answers Null hypothesis: –“There is no difference in mean number of fights between inmates with tattoos and inmates without tattoos.” Use a 1 or 2-tailed test? –One-tailed test because the theory predicts that inmates with tattoos will get into MORE fights.

Answers Calculations –Obtained value Reject the null? –Yes because the t(obtained) (19.09) is greater than the t(critical, one-tail, df=398) (1.658) This t value indicates there are standard error units that separate the two mean values –VERY unlikely we got this big a difference due to sampling error Research hypothesis restated as non-directional: –“There is a difference in the mean number of fights reported by inmates with tattoos and inmates without tattoos.” Would you come to a different conclusion if you used a 2-tailed test? –No, because is still well beyond the 2-tailed critical value (1.980).

2-Sample Hypothesis Testing in SPSS Independent Samples t Test Output: –Testing the H o that there is no difference in number of adult arrests between a sample of individuals who were abused/neglected as children and a matched control group.

Interpreting SPSS Output Difference in mean # of adult arrests between those who were abused as children & control group

Interpreting SPSS Output t statistic, with degrees of freedom

Interpreting SPSS Output “Sig. (2 tailed)” –gives the actual probability of making a Type I (alpha) error a.k.a. the “p value” – p = probability

“Sig.” & Probability Number under “Sig.” column is the exact probability of obtaining that t-value (finding that mean difference) if the null is true –When probability > alpha, we do NOT reject H 0 –When probability < alpha, we DO reject H 0 As the test statistics (here, “t”) increase, they indicate larger differences between our obtained finding and what is expected under null –Therefore, as the test statistic increases, the probability associated with it decreases

Example 2: Education & Age at which First Child is Born H 0 : There is no relationship between whether an individual has a college degree and his or her age when their first child is born.

Education & Age at which First Child is Born 1.What is the mean difference in age? 2.What is the probability that this t statistic is due to sampling error? 3.Do we reject H 0 at the alpha =.05 level? 4.Do we reject H 0 at the alpha =.01 level?