Statistical Inference: A Review of Chapters 12 and 13

1. Statistical Inference:A Review of Chapters 12 and 13 Chapter 14

2. 14.1 Introduction • In this chapter we build a framework that helps decide which technique (or techniques) should be used in solving a problem. • Logical flow chart of techniques for Chapters 12 and 13 is presented next.

3. Describe a population Compare two populations Nominal Interval Nominal Interval Z test & estimator of p Variability Central location Variability Central location t- test & estimator of m c2- test & estimator of s2 Continue Continue Summary Problem objective? Data type? Data type? Z test & estimator of p1-p2 Type of descriptive measurement? Type of descriptive measurement? Experimental design? F- test & estimator of s12/s22

4. Experimental design? Independent samples Matched pairs Population variances? Equal Unequal t- test & estimator of m1-m2 (Unequal variances) t- test & estimator of m1-m2 (Equal variances) t- test & estimator of mD Continue Continue

5. Identifying the appropriate technique • Example 14.1 • Is the antilock braking system (ABS) really effective? • Two aspects of the effectiveness were examined: • The number of accidents. • Cost of repair when accidents do occur. • An experiment was conducted as follows: • 500 cars with ABS and 500 cars without ABS were randomly selected. • For each car it was recorded whether the car was involved in an accident. • If a car was involved with an accident, the cost of repair was recorded.

6. Identifying the appropriate technique • Example – continued • Data • 42 cars without ABS had an accident, • 38 cars equipped with ABS had an accident • The costs of repairs were recorded (see Xm14-01). • Can we conclude that ABS is effective?

7. Identifying the appropriate technique • Solution • Question 1: Is there sufficient evidence to infer that the accident rate is lower in ABS equipped cars than in cars without ABS? • Question 2: 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? • Question 3: How much cheaper is it to repair ABS equipped cars than cars without ABS?

8. Question 1: Compare the accident rates • Solution – continued Problem objective? Describing a single population Compare two populations Data type? A car had an accident: Yes / No Nominal Interval Z test & estimator of p1-p2

9. Use case 1 test statistic Question 1: Compare the accident rates • Solution – continued • p1 = proportion of cars without ABS involved with an accidentp2 = proportion of cars with ABS involved with an accident • The hypotheses testH0: p1 – p2 = 0H1: p1 – p2 > 0

10. Do not reject H0. Question 1: Compare the accident rates • Solution – continued • Use Test Statistics workbook: z-Test_2 Proportions(Case 1) worksheet 42/500 38/500

11. Question 2: Compare the mean repair costs per accident • Solution - continued Problem objective? Describing a single population Compare two populations Data type? Cost of repair per accident Nominal Interval Type of descriptive measurements? Variability Central location

12. Population variances equal? Equal t- test & estimator of m1-m2 (Equal variances) Question 2: Compare the mean repair costs per accident Central location • Solution - continued Experimental design? Independent samples Matched pairs Run the F test for the ratio of two variances. Equal Unequal

13. Question 2: Compare the mean repair costs per accident • Solution – continued • m1 = mean cost of repairing cars without ABSm2 = mean cost of repairing cars with ABS • The hypotheses tested H0: m1 – m2 = 0 H1: m1 – m2 > 0 • For the equal variance case we use

14. Do not reject H0. There is insufficient evidence to conclude that the two variances are unequal. Question 2: Compare the mean repair costs per accident • Solution – continued • To determine whether the population variances differ we apply the F test • From Excel Data Analysis we have (Xm14-01)

15. Question 2: Compare the mean repair costs per accident • Solution – continued • Assuming the variances are really equal we run the equal-variances t-test of the difference between two means At 5% significance levelthere is sufficient evidenceto infer that the cost of repairsafter accidents for cars with ABS is smaller than the cost of repairs for cars without ABS.

16. Checking required conditions • The two populations should be normal (or at least not extremely nonnormal)

17. Question 3: Estimate the difference in repair costs • Solution • Use Estimators Workbook: t-Test_2 Means (Eq-Var) worksheet We estimate that the cost of repairing a car not equipped with ABS is between $71 and $651 more expensive than to repair an ABS equipped car.