Practice

2. Practice • For an SAT test •  = 500 •  = 100 • What is the probability that a single person will have a SAT score of 550 or higher?

3. Step 1: Sketch out question -3 -2 -1 0 1 2 3

4. Step 2: Calculate the Z score (550 - 500) / 100 = .50 -3 -2 -1 0 1 2 3

5. Step 3: Look up Z score in Table Z = .50; Column C =.3085 .3085 -3 -2 -1 0 1 2 3

6. Practice • There is a .3085 probability that any single person will have an SAT score of 550 or higher.

7. Practice • For an SAT test •  = 500 •  = 100 • What is the probability that a group of 10 people have a mean SAT score of 550 or higher?

8. Step 1: Sketch out question -3 -2 -1 0 1 2 3

9. Step 2: Calculate the standard error 100 / √10 = 31.62 -3 -2 -1 0 1 2 3

10. Step 3: Calculate the Z score (550 - 500) / 31.62 = 1.58 -3 -2 -1 0 1 2 3

11. Step 4: Look up Z score in Table Z = 1.58; Column C =.0571 .0571 -3 -2 -1 0 1 2 3

12. The next step • Based on a sample of participants how accurate is a given sample mean at estimating the population mean? • Example: • Opinion polls

13. Confidence Intervals • Using a sample of US citizens you ask if they approve of the president. • You find 43% approve • “The findings of the survey are accurate within 3 points, using a 95% level of confidence”

14. Confidence Intervals • Using a sample of US citizens you ask if they approve of the president. • You find 43% approve • Thus you are 95% confident the population values falls between 40% to 46%

15. Confidence Intervals • A recent survey found that on average a person each 10 pieces of fruit a week. • The findings of the survey are accurate within 5 pieces of fruit, using a 95% level of confidence • What does this mean?

16. Confidence Intervals • A recent survey found that on average a person each 10 pieces of fruit a week. • The findings of the survey are accurate within 5 pieces of fruit, using a 95% level of confidence • 95% confident that on average a person eats between 5 and 15 pieces of fruit a week.

17. Note • You can not use the Z formula to answer this type of question • You usually don’t know the  or the  • If you did you wouldn’t need a sample! • Better to use a “t distribution”

18. t distribution • Different sample sizes have different t distributions • When using t distributions you must know the degrees of freedom (df) • df = N - 1 • Thus, the df can range from 1 to 

19. t distribution

20. t distribution • There are an infinite number of t distributions; however, once df is greater than 120 it doesn’t change much • Table D page 392 • First Column are df • For this chapter you will look at the top row

21. t distribution 95% 2.5% 2.5% -1.96 1.96 df = 

22. t distribution 95% 2.5% 2.5% -1.96 1.96 df =  NOTICE: Similarities to the z table

23. t distribution 95% 2.5% 2.5% -2.26 2.26 df = 9

24. t distribution 95% 2.5% 2.5% -2.26 2.26 df = 9

25. t distribution 95% 2.5% 2.5% -4.30 4.30 df = 2

26. t distribution 95% 2.5% 2.5% -4.30 4.30 df = 2

27. Confidence Intervals Around a Mean • Inferential statistic • Based on a sample you calculate a mean • With a certain degree of confidence (usually 95%) you can state that the population mean is in the interval

28. Confidence Intervals • To find the lower and upper limits: LL = X - t (sx) UL = X + t(sx) Where: X is the mean of the sample t is a value from the t distribution table sx is the standard error of the mean, calculated from a sample

29. Confidence Intervals • Note: Similar to the backwards z LL = X - t (sx) UL = X + t(sx) X =  + (z)()

30. Confidence Intervals • To find the lower and upper limits: LL = X - t (sx) UL = X + t(sx) Where: X is the mean of the sample t is a value from the t distribution table sx is the standard error of the mean, calculated from a sample

31. Confidence Intervals Sx = S / N

32. Remember S = -1

33. Example • You are curious about how many hours a week students study. You sample 5 students: • 20, 22, 19, 23, 17 • What is the mean score? What are its 95% confidence intervals?

34. Step 1: Calculate S

35. Step 1: Calculate S NOTE: In other examples you may be given X and X2. Make sure you understand where they came from!

36. 101 2.39 = 2063 5 5 - 1 N = 5 X = 101  X2 = 2063

37. Step 2: Calculate Sx Sx = S / N 1.07 = 2.39 / 5

38. Step 3: Look up t value • df = N - 1 • 4 = 5 - 1 • Want 95% confidence • t = 2.776

39. Step 4: Calculate LL and UL LL = X - t (sx) UL = X + t(sx) X = 20.2 Sx = 1.07 t = 2.776

40. Step 4: Calculate LL and UL LL = 20.2 - 2.776 (1.07) = 17.23 UL = 20.2 + 2.776(1.07) = 23.17 X = 20.2 Sx = 1.07 t = 2.776

41. Example • Thus, you are 95% confident that the average student studies between 17.23 and 23.17 hours a week

42. Example • Using the same data -- what if you want to be 99.9% confident.

43. Step 3: Look up t value • df = N - 1 • 4 = 5 - 1 • Want 99.9% confidence • t = 8.610

44. Step 4: Calculate LL and UL LL = X - t (sx) UL = X + t(sx) X = 20.2 Sx = 1.07 t = 8.610

45. Step 4: Calculate LL and UL LL = 20.2 - 8.610 (1.07) = 10.99 UL = 20.2 + 8.610(1.07) = 29.41 X = 20.2 Sx = 1.07 t = 2.776

46. Example • Thus, you are 95% confident that the average student studies between 17.23 and 23.17 hours a week • You are 99.9% confident that the average student studies between 10.99 and 29.41 hours a week. • As you get more confident your intervals will get larger

47. Practice Page 168 8.24

48. Practice • M = 1.20 • S hat = .10 • SE = .0577 • t(2) = 31.598 • LL = -.62 • UL = 3.02 • You can be 99.9% confident that the average burn rate of pots in this box is below 4 inches per minute.