1. 366_8

2. Estimation: Chapter 8 Suppose we observe something in a random sample how confident are we in saying our observation is an accurate reflection of the population?

3. Estimation Confidence intervals – the range in which the population parameter is estimated to be – ‘margin of error’ – accounting for sampling error

4. Estimation Confidence intervals – We need: standard error (we calculate this) the estimated mean a choice of confidence level – 68% – 90% – 95% – 99%

5. Estimation Confidence intervals – We need: mean (Spss gets this) Z value for confidence level (we pick this) standard error of the mean (we calculate this) – Calculated with standard deviation and sample size – Larger sample, less error – Smaller standard deviation, less error

6. Estimation Confidence intervals – We need: standard error of the mean – s.e. = standard deviation / sqrt of N – Example 500 students have mean of 7.5 hrs/week of commute time, std. deviation = 1.5 hrs s.e. = 1.5 / sqrt of 500 =.067

7. Estimation Confidence intervals – We need: standard error of the mean – s.e. = standard deviation / sqrt of N – Changed example 300 students have mean of 7.5 hrs/week of commute time, std. deviation = 3.5 hrs s.e. = 3.5 / sqrt of 300 =.20

8. Estimation Confidence intervals – We need: Select confidence interval (68%, 90%, 95%, 99%) 68% CI = mean +/- 1 (s.e) 90% CI = mean +/- 1.64 (s.e) 95% CI = mean +/- 1.96 (s.e) 99% CI = mean +/- 2.58 (s.e)

9. Estimation Confidence intervals – 95% confidence: CI=7.5hrs +/- 1.96 (.07) = 7.5hrs +/- 0.14 hrs = 7.36 to 7.64 hrs We are 95% confidence that population mean is between 7.36 and 7.64

10. Estimation Confidence intervals – 68% confidence: CI=7.5hrs +/- 1 (.07) = 7.5hrs +/- 0.07 hrs = 7.43 to 7.57 hrs We are 68% confidence that population mean is between 7.43 and 7.57

11. Estimation Confidence intervals – =99% confidence: CI=7.5hrs +/- 2.58 (.07) = 7.5hrs +/- 0.18 hrs = 7.43 to 7.57 hrs We are 99% confidence that population mean is between 7.32 and 7.68

12. Estimation At any level of confidence – The interval is determined by sample size and the standard deviation of the estimated mean – More variation around mean, less confident – Fewer observations, less confident

13. Estimation So far, we had interval data (Hours of commute) Works different if nominal – Approve or disapprove of Obama – A proportion (percent), not a mean – Different formula

14. Estimation Confidence interval for proportion – We need Standard error (we calculate, again) observed proportion select our confidence level

15. Estimation Confidence interval for proportion – We need Standard error s.e.p. = sqrt [of (p)*(1-p) / n] CI = p +/- Z (s.e.p) Obama estimated at.46 approval in Pew Values Survey CI =.46 +/- 1.96 (s.e.p)

16. Estimation Confidence interval for proportion – We need Standard error s.e.p. = sqrt [of (.46)*(1-.46) / n ] = sqrt of ((.46)*(1-.46) / 1515 ) = sqrt of (.2484 / 1514) =.013 Obama estimated at.46 approval in Pew Values Survey CI =.46 +/- 1.96 (.013):

17. Estimation Confidence interval for proportion – Obama estimated at.46 approval in Pew Values Survey CI =.46 +/- 1.96 (.013) =.46 +/-.025 95% confident population approval of Obama is between.435 and.485 or between 43.5% and 48.5%

18. Hypothesis Testing: Chpt 9 Statistics test a Null Hypothesis The mean age for tea party supporters and non supporters is the same There is no difference between tea party supporters and non supporters

19. Hypothesis Testing: Chpt 9 Statistics test a Null Hypothesis Support for the Tea Party is independent of gender Gender does not affect support for the Tea Party

20. Hypothesis Testing: Chpt 9 Statistical significance – Probability that the NULL is wrong – Probability that nothing is going on – Probability that an observed relationship is a sampling fluke

21. Hypothesis Testing: Chpt 9 Statistical significance – We need to decide what is ‘significantly improbable’ – The level we reject the null hypothesis happens just 5% of the time? (.05 alpha) just 1% of the time (.01 alpha)

22. Statistical significance Type I vs Type II Errors decisionNull is trueNull is false Reject nullType I errorcorrect decision Retain nullCorrect decisionType II error Significance (alpha) is chance of a Type I error We want to avoid Type I errors, Type II are less dangerous: Drug trials, criminal justice

23. Hypothesis Testing with t-test Research Hypothesis (H1): Something is going on. There is a difference between groups, Men have higher score. H1: X m > X f Null Hypothesis (H0): There is no difference Mean for group 1 = the mean for group 2 H0: X1 = X2

24. Hypothesis Testing with t Observe difference between means: Magnitude of difference Variance in measure of X1 and X2 Number of observations What is the likelihood that such a difference would occur by chance?

25. T-test Assume – Random samples, independent of each other – Variable being compared is interval or ratio – Distributions are normal – Roughly equal variance of each group

26. T-test Decide criteria, or critical t – Alpha to reject (chance of a Type 1 error) t= 1.65 for alpha =.10 t= 1.96 for alpha =.05 – Directional test? do you think value is higher/lower for a specific group?

27. t test Calculate t t = mean1 – mean 2 ________________ s x1-x2  ----------std. error of the difference between 2 means this part is messy, but includes info about sample sizes and variances of each mean

28. t-test result is one value (a t-statistic) we can use to check if difference between groups is significant Example: – Corruption, south vs. non south What hypothesis?

29. Projects Identify testable hypotheses – x causes y – x explains differences in y – differences in x explain y – x and y go together in some interesting way State null hypothesis

30. Mean of group 1 significantly different than mean of group 2? Non south, x=.33; s.e..03South, x=.43; s.e.05

31. t-test Southern states, zoomed in....

32. Results ttest percap_convic, by(var82) Two-sample t test with equal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- 0 | 39.3358051.0313071.1955129.2724272.3991831 1 | 11.4324917.0577252.1914528.303872.5611115 ---------+-------------------------------------------------------------------- combined | 50.3570762.0278429.1968793.3011237.4130287 ---------+-------------------------------------------------------------------- diff | -.0966866.0664606 -.2303146.0369414 ------------------------------------------------------------------------------ diff = mean(0) - mean(1) t = -1.4548 Ho: diff = 0 degrees of freedom = 48 Ha: diff 0 Pr(T |t|) = 0.1522 Pr(T > t) = 0.9239.

33. SPSS Results: Age * gender

34. SPSS results

35. Results Note each mean is given variation around mean is given confidence intervals difference between means is given(-.096) std. error of differences btwn means given AND t values

36. Results Note different t values are given Each is for a specific hypothesis – Difference is greater than 0, positive(one tail) – Difference is greater than 0, negative(one tail) – “Absolute difference” (two tail)

37. t test results Do we accept of reject null hypothesis?