Chapter 14

1. Chapter 14 Statistical Inference: Review of Chapters 12 & 13 Sir Naseer Shahzada

2. Which technique to use?

3. Identifying the Correct Technique… • The two most important factors in determining the correct statistical technique to use are: • the problem objective, (i.e. describeone population or comparetwo populations) • and the data type. (i.e. interval data or nominal data) Once these factors are determined, our analysis extends to other factors (e.g. type of descriptive measure [central location? variability?], etc.)

4. Figure 14.1… • The flowchart in your textbook (Figure 14.1) describes the logical process that allows us to identify the appropriate method to use for the problem. • Start at the top and work • your way down the chart… • The following slides are • an interactive version of • this flowchart…

5. Figure 14.1: Flowchart of Techniques… Problem objective? Describe a population Compare two populations Click on the mouse icon to follow the branch of the flowchart to the next level… Skip flowchart, go to examples…

6. Figure 14.1: Flowchart of Techniques… Problem objective? Describe a population Data type? Nominal Interval

7. Variability Central location Figure 14.1: Flowchart of Techniques… Problem objective? Describe a population Data type? Type of descriptive measurement? Interval

8. Slide 12.19 : Identifying Factors… Top of Flowchart • Factors that identify the t-test and estimator of :

9. Slide 12.29 : Identifying Factors… Top of Flowchart • Factors that identify the chi-squared test and • estimator of :

10. Figure 14.1: Flowchart of Techniques… Problem objective? Describe a population Data type? Nominal

11. Slide 12.42 : Identifying Factors… Top of Flowchart • Factors that identify the z-test and • interval estimator of p:

12. Nominal Interval Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type?

13. Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Nominal

14. Slide 13.64 : Identifying Factors… Top of Flowchart • Factors that identify the z-test • and estimator for p1–p2

15. Variability Central location Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Interval Type of descriptive measurement?

16. Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Interval Type of descriptive measurement? Variability

17. Slide 13.49 : Identifying Factors… Top of Flowchart • Factors that identify the F-test and • estimator of :

18. Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Type of descriptive measurement? Central Location Interval Experimental design? Matched pairs Independent samples

19. Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Type of descriptive measurement? Central Location Interval Experimental design? Matched pairs

20. Slide 13.40 : Identifying Factors… Top of Flowchart • Factors that identify the t-test and estimator of :

21. Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Type of descriptive measurement? Central Location Interval Experimental design? Population variances? Independent samples Equal Unequal

22. Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Type of descriptive measurement? Central Location Interval Experimental design? Population variances? Independent samples Equal

23. Slide 13.30 : Identifying Factors… Top of Flowchart • Factors that identify the equal-variances t-test • and estimator of :

24. Figure 14.1: Flowchart of Techniques… Problem objective? Compare two populations Data type? Type of descriptive measurement? Central Location Interval Experimental design? Population variances? Independent samples Unequal

25. Slide 13.31 : Identifying Factors… Top of Flowchart • Factors that identify the unequal-variances t-test and estimator of :

26. Example 14.1… • Is anti-lock braking system (ABS) in cars really effective? • We would expect if it were effective that: •  The number of accidents would decrease, and •  The cost of accident repairs would be less. • Data were collectedon 500 cars with ABS and 500 cars without. The number of cars involved in accidents was recorded, as was the cost of repairs. What can we conclude?

27. Example 14.1 (a)… IDENTIFY • Is there sufficient evidence to infer that the accident rate is lower in ABS-equipped cars than in cars without ABS? • (If ABS is effective, we would expect a lower accident rate in ABS-equipped cars.) • Accident rate = number of cars in accidents • total number of cars • This is nominal (i.e. categorical) data; either a car had an accident or it didn’t. The accident rate is a proportion. We want to compare cars with ABS against cars without.

28. Example 14.1 (a)… IDENTIFY • Identify the correct technique… Problem objective? Describing a single population Compare two populations Data type? Nominal Interval

29. Example 14.1 (a)… IDENTIFY • The correct technique: • has been identified. The next step is to translate the rest of the problem into the symbols and language of statistics: • p1 = proportion of cars without ABS involved in an accident • p2 = proportion of cars with ABS involved in an accident • We want to test if ABS is effective, that is, we want to research if: p1 > p2 , that is if H1: (p1 – p2 ) > 0

30. Example 14.1 (a)… COMPUTE • Since H1: (p1 – p2 ) > 0 • we have our null hypothesis: H0: (p1 – p2 ) = 0 • Hence this is a Case 1 type problem. • Upon calculating our sample proportions… • …we can use Excel to complete our analysis…

31. Example 14.1 (a)… INTERPRET • Noting that z = .4663 is not greater than zCritical = 1.6449 (or alternatively, by looking at the p-value of .3205), we cannot reject H0 in favor of H1, that is, there is not enough evidence to infer that ABS equipped cars have fewer accidents than non-ABS equipped cars…

32. Example 14.1 (b)… IDENTIFY • When accidents do occur, we would expect the severity of accidents to be lower in ABS-equipped cars (assuming that ABS is effective), thus we are interested in this question: • Is there sufficient evidence to infer that the cost of repairing accident damage in ABS-equipped cars is less than that of cars without ABS? • The cost of repairs is interval data. We need a measure to compare the two populations of cars in a meaningful way…

33. Example 14.1 (b)… IDENTIFY • Identify the correct technique… Problem objective? Describing a single population Compare two populations Data type? Nominal Interval Type of descriptive measurements? Variability Central location …continues…

34. Population variances equal? Example 14.1 (b)… IDENTIFY • Identify the correct technique (continued)… Central location Experimental design? Independent samples Matched pairs Equal?? Unequal?? e.g. we are not comparing a head-on collision of an ABS equipped car with a head-on collision of a non-ABS car… Which one is it?

35. Population variances equal? Example 14.1 (b)… IDENTIFY • Identify the correct technique (continued)… Apply the F-test of Unequal?? Equal?? there is not enough evidence to infer that the variances differ…

36. Population variances equal? Example 14.1 (b)… IDENTIFY • Identify the correct technique (continued)… • …we have the right technique! Let’s proceed with our hypotheses set-up… Unequal Equal

37. Example 14.1 (b)… IDENTIFY • We want to research whether or not the mean cost of repairing cars without ABS brakes (population 1) is greater than the mean cost of repair of cars equipped with ABS brakes (population 2), i.e.:

38. Example 14.1 (b)… INTERPRET • Applying the Data Analysis tools in Excel to our data… • Indeed, there is sufficient evidence to support the belief that non-ABS equipped cars do indeed have higher accident repair costs than ABS equipped cars.

39. Example 14.1 (c)… IDENTIFY • In part (b) we’ve shown that ABS-equipped cars suffer less damage in accidents (as measured by repair costs); can we estimate how much cheaper they are to repair on average compared to cars without ABS brakes. • Our path through the flowchart is the same, that is, we are comparing the measure of central location of independent samples of interval data from two populations who’s variances are equal…

40. Example 14.1 (c)… CALCULATE • The estimator of interest is: • Assuming a 95% confidence interval… • …we estimate the cost of repair for a non-ABS equipped car of between $71 and $651 over an ABS equipped car.