1. The Normal Distribution

2. n = 20,290  = 2622.0  = 2037.9 Population

3. Y = 2675.4 s = 1539.2 Y = 2588.8 s = 1620.5 Y = 2702.4 s = 1727.1 Y = 2767.2 s = 2044.7 SAMPLES

4. Y = 2675.4 s = 1539.2 Y = 2588.8 s = 1620.5 Y = 2702.4 s = 1727.1 Y = 2767.2 s = 2044.7 Sampling distribution of the mean

5. 1000 samples Sampling distribution of the mean

7. Non-normal Approximately normal

9. Sample means are normally distributed The mean of the sample means is . The standard deviation of the sample means is *If the variable itself is normally distributed, or sample size (n) is large

10. Standard error The standard error of an estimate of a mean is the standard deviation of the distribution of sample means We can approximate this by

11. Distribution of means of samples with n =10

12. Larger samples equal smaller standard errors

13. Central limit theorem

14. Button pushing times Frequency Time (ms)

15. Distribution of means

16. Binomial Distribution

17. Normal approximation to the binomial distribution

18. Example A scientist wants to determine if a loonie is a fair coin. He carries out an experiment where he flips the coin 1,000,000 times, and counts the number of heads. Heads come up 543,123 times. Using these data, test the fairness of the loonie.

19. Inference about means Because is normally distributed:

20. But... We don’t know  A good approximation to the standard normal is then: Because we estimated s, t is not exactly a standard normal!

21. t has a Student’s t distribution }

22. Degrees of freedom df = n - 1 Degrees of freedom for the student’s t distribution for a sample mean:

23. Confidence interval for a mean

24.  (2) = 2-tailed significance level Df = degrees of freedom SE Y = standard error of the mean

25. 95% confidence interval for a mean Example: Paradise flying snakes 0.9, 1.4, 1.2, 1.2, 1.3, 2.0, 1.4, 1.6 Undulation rates (in Hz)

26. Estimate the mean and standard deviation

27. Find the standard error

28. Table A3.3

29. Find the critical value of t

30. Putting it all together...

31. 99% confidence interval

32. Confidence interval for the variance

34. Table A3.1

35. 95% confidence interval for the variance of flying snake undulation rate

36. 95% confidence interval for the standard deviation of flying snake undulation rate

37. One-sample t-test

38. Hypotheses for one-sample t-tests H 0 : The mean of the population is  0. H A : The mean of the population is not  0.

39. Test statistic for one-sample t-test  0 is the mean value proposed by H 0

40. Example: Human body temperature H 0 : Mean healthy human body temperature is 98.6ºF H A : Mean healthy human body temperature is not 98.6ºF

41. Human body temperature

42. Degrees of freedom df = n-1 = 23

43. Comparing t to its distribution to find the P-value

44. A portion of the t table

46. -1.67 is closer to 0 than -2.07, so P >  With these data, we cannot reject the null hypothesis that the mean human body temperature is 98.6.

47. Body temperature revisited: n = 130

48. t is further out in the tail than the critical value, so we could reject the null hypothesis. Human body temperature is not 98.6ºF.

49. One-sample t-test: Assumptions The variable is normally distributed. The sample is a random sample.