The Two Sample t Review significance testing Review

The Two Sample t Review significance testing Review t distribution Introduce 2 Sample t test / SPSS

Significance Testing • State a Null Hypothesis • Calculate the odds of obtaining your sample finding if the null hypothesis is correct – Compare this to the odds that you set ahead of time (e. g. , alpha) – If odds are less than alpha, reject the null in favor of the research hypothesis • The sample finding would be so rare if the null is true that it makes more sense to reject the null hypothesis

Significance the old fashioned way • Find the “critical value” of the test statistic for your sample outcome – Z tests always have the same critical values for given alpha values (e. g. , . 05 alpha +/- 1. 96) • Use if N >100 – t values change with sample size • Use if N < 100 • As N reaches 100, t and z values become almost identical • Compare the critical value with the obtained value Are the odds of this sample outcome less than 5% (or 1% if alpha =. 01)?

Critical Values/Region for the z test ( =. 05)

Directionality • Research hypothesis must be directional – Predict how the IV will relate to the DV • Males are more likely than females to… • Southern states should have lower scores…

Non-Directional & Directional Hypotheses • Nondirectional – Ho: 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 -1. 96 • Directional – H 1: sample mean is larger than population mean (X > µ) – Ho x ≤ µ – APPLY 1 -TAILED TEST • 5% chance of error in one tail 1. 65

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 Practice: ALPHA. 05. 01. 10. 05 TEST 2 -tailed 1 -tailed N 57 25 32 15 t(Critical)

Example: Single sample means, smaller N 1. A random sample of 16 UMD students completed an IQ test. They scored an average of 104, with a standard deviation of 9. The IQ test has a national average of 100. IS the UMD students average different form the national average? 2. Professor Fred hypothesizes that Duluthians are more polite than the average American. A random sample of 50 Duluth residents yields an average score on the Fred politeness scale (FPS) of 31 (s = 9). The national average is known to be 29. Write the research an null hypotheses. What can you conclude? • USE ALPHA =. 05 for both

Answer #1 Conceptually • Under the null hypothesis (no difference between means), there is more than a 5% chance of obtaining a mean difference this large. t (obtained) = 1. 72 Critical Region Sampling distribution for one sample t-test (a hypothetical plot of an infinite number of mean differences, assuming null was correct) Critical Region -2. 131 (t-crit, df=15)

“ 2 -Sample” t test – 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 (Ho) • 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 2 -Sample Sampling Distribution – difference between sample means (closer sample means will have differences closer to 0) • t-test for the difference between 2 sample means: • Does our observed difference between the sample means reflects a real difference in the population means or is due to sampling error? - t critical ASSUMING THE NULL! 0 t critical

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=2. 0 » Non soc. majors: N = 105, mean=15, s=2. 5 » Alpha =. 01 » FORMULA: t(obtained) = X 1 – X 2 pooled estimate

Answers • Null Hypothesis (Ho): Soc. majors at UMD DO NOT drink more beers per month than non-soc. Majors • Directional 1 tailed test • tcritical for 1 tailed test with 203 df and =. 05 (use infinity) = 1. 645 • tobtained = 3. 13 – Reject null hypothesis: there is less than a 1% chance of obtaining a mean difference of 1 beer if the null hypothesis is true. • 3. 13 standard errors separates soc and non-soc majors with respect to their average beer consumption

Example 2 • Dr. Phil believes that inmates with tattoos will get in more fights than inmates without tattoos. • Tattooed inmates N = 25, s = 1. 06, mean = 1. 00 • Non-Tattooed inmates N = 37, s =. 5599, mean = 0. 5278 – Null hypothesis? – Directional or non? – tcritical? – Difference between means? – Significant at the. 01 level?

Answers • Null hypothesis: – Inmates with tattoos will NOT get in more fights than inmates without tattoos. • Use a 1 or 2 -tailed test? – One-tailed test because theory predicts that inmates with tattoos will get into MORE fights. • tcritical (60 df) = 2. 390 • tobtained = 2. 23

Answers • Reject the null? – No, because the t(obtained) (2. 23) is less than the t(critical, one-tail, df=398) (1. 658) • This t value indicates there are 2. 23 standard error units that separate the two mean values – Greater than a 1% chance of getting this finding under null • SPSS DEMO

2 -Sample Hypothesis Testing in SPSS • Independent Samples t Test Output: – Testing the Ho 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 obtaining this finding if the null is correct • a. k. a. the “p value” – p = probability • The odds are NOT ZERO (if you get. ooo, interpret as <. 001)

“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?