Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Fall 2015 Room 150 Harvill Building 10: :50 Mondays, Wednesdays & Fridays.

By the end of lecture today 10/28/15 Logic of hypothesis testing Steps for hypothesis testing Levels of significance (Levels of alpha) what does p < 0.05 mean? what does p < 0.01 mean? One-sample z-tests and t-tests

Before next exam (November 20 th ) Please read chapters in OpenStax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

Homework Assignment Worksheet Assignment 16 One-sample z and t hypothesis tests Due: Friday, October 30 th

Everyone will want to be enrolled in one of the lab sessions Labs continue this week

Confidence Interval of 99% Has and alpha of 1% α =.01 Confidence Interval of 90% Has and alpha of 10% α =. 10 Confidence Interval of 95% Has and alpha of 5% α =.05 99%95%90% Area outside confidence interval is alpha Area in the tails is called alpha Area associated with most extreme scores is called alpha Critical z Critical z 2.58 Critical z Critical z 1.96 Critical z Critical z 1.64

Rejecting the null hypothesis The result is “statistically significant” if: the observed statistic is larger than the critical statistic (which can be a ‘z” or “t” or “r” or “F” or x 2 ) observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x 2 ) to be big!! the p value is less than 0.05 (which is our alpha) p < 0.05 If we want to reject the null, we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis

How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level = or or What if our observed z = 2.0? Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels p < 0.05 Yes, Significant difference Not a Significant difference

How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level = or or What if our observed z = 1.5? Do Not Reject the null Do Not Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels Not a Significant difference

How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level = or or What if our observed z = -3.9? Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels p < 0.05 Yes, Significant difference p < 0.01 Yes, Significant difference

How would the critical z change? α = 0.05 Significance level =.05 α = 0.01 Significance level = or or What if our observed z = -2.52? Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z Deciding whether or not to reject the null hypothesis.05 versus.01 alpha levels p < 0.05 Yes, Significant difference Not a Significant difference

z score = 1.64 One versus two tail test of significance: Comparing different critical scores (but same alpha level – e.g. alpha = 5%) One versus two tailed test of significance One-tailed test: If your hypothesis is “directional” claiming that one group will have a bigger mean than the other group Two-tailed test: If your hypothesis is “non-directional” claiming only that the two groups have different means How would the critical z change? Pros and cons… 5% 95% 2.5% 95% 2.5%

One versus two tail test of significance 5% versus 1% alpha levels or How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level = or or or % 2.5% 1%.5%

-1.64 or How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level = or or or What if our observed z = 2.0? Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z One versus two tail test of significance 5% versus 1% alpha levels

-1.64 or How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level = or or or What if our observed z = 1.75? Reject the null Do not Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z One versus two tail test of significance 5% versus 1% alpha levels

-1.64 or How would the critical z change? One-tailedTwo-tailed α = 0.05 Significance level =.05 α = 0.01 Significance level = or or or What if our observed z = 2.45? Reject the null Do not Reject the null Remember, reject the null if the observed z is bigger than the critical z One versus two tail test of significance 5% versus 1% alpha levels

Standard deviation and Variance For Sample and Population These would be helpful to know by heart – please memorize these formula Review

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem One or two tailed test? Balance between Type I versus Type II error Critical statistic (e.g. z or t or F or r) value?

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem Critical statistic (e.g. z or t) value? How is a t score same as a z score? Population versus sample standard deviation How is a t score different than a z score?

Hypothesis testing: one sample t-test Is the mean of my observed sample consistent with the known population mean or did it come from some other distribution? We are given the following problem: 800 students took a chemistry exam. Accidentally, 25 students got an additional ten minutes. Did this extra time make a significant difference in the scores? The average number correct by the large class was 74. The scores for the sample of 25 was 76, 72, 78, 80, 73 70, 81, 75, 79, 76 77, 79, 81, 74, 62 95, 81, 69, 84, 76 75, 77, 74, 72, 75 Please note: In this example we are comparing our sample mean with the population mean (One-sample t-test) L e t ’ s j u m p r i g h t i n a n d d o a t - t e s t

Hypothesis testing Step 1: Identify the research problem / hypothesis Did the extra time given to this sample of students affect their chemistry test scores? Null: The extra time did not affect their chemistry test scores Alternative: The extra time did affect their chemistry test scores Independent Variable? Dependent Variable? IV: Nominal Ordinal Interval or Ratio? DV: Nominal Ordinal Interval or Ratio? Extra time vs. no extra time Test Scores IV: Nominal DV: Ratio One tail or two tail test? Two-tailed test

Hypothesis testing Step 2: Decision rule =.05 Degrees of freedom (df) = ( n - 1) = (25 - 1) = 24 two tail test n = 25 What is formula for degres of freedom? n – 1

α =.05 (df) = 24 Critical t (24) = two tail test

. Hypothesis testing Step 3: Calculations µ = 74 = – – – – – – – – – – – – – – – – – – – – – – – – – = = = = = = = = = = = = = = = = = = = = = = = = = x (x - x) (x - x) = 6.01 N = 25 Σx = 1911 Σ(x- x) = 0 Σ(x- x) 2 = ΣxΣx = N =

. s = 6.01 = = critical t Observed t (24) = 2.03 Step 3: Calculations µ = 74 = N = 25 ΣxΣx = N = Hypothesis testing

. Step 4: Make decision whether or not to reject null hypothesis Step 6: Conclusion: The extra time did not have a significant effect on the scores 2.03 is not farther out on the curve than 2.064, so, we do not reject the null hypothesis Observed t (24) = 2.03 Critical t (24) = 2.064

. Hypothesis testing: Did the extra time given to these 25 students affect their average test score? notice we are comparing a sample mean with a population mean: single sample t-test The mean score for those students who where given extra time was percent correct, while the mean score for the rest of the class was only 74 percent correct. A t-test was completed and there appears to be no significant difference in the test scores for these two groups t(24) = 2.03; n.s. Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus z-test) with brief overview of results Finish with statistical summary t(24) = 2.03; ns Or if it had been different results that *were* significant: t(24) = -5.71; p < 0.05 Type of test with degrees of freedom n.s. = “not significant” p<0.05 = “significant” n.s. = “not significant” p<0.05 = “significant” Value of observed statistic