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

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 Problem objective? Describe a populationCompare two populations Data type? Interval Nominal Interval Nominal Type of descriptive measurement? Type of descriptive measurement? Z test & estimator of p Z test & estimator of p Z test & estimator of p 1 -p 2 Z test & estimator of p 1 -p 2 Central location Variability Central location Variability t- test & estimator of  t- test & estimator of    - test & estimator of  2   - test & estimator of  2 F- test & estimator of   2 /   2 F- test & estimator of   2 /   2 Experimental design? Continue Summary

4 Continue t- test & estimator of  1 -  2 (Unequal variances) t- test & estimator of  1 -  2 (Unequal variances) Population variances? t- test & estimator of  D t- test & estimator of  D t- test & estimator of  1 -  2 (Equal variances) t- test & estimator of  1 -  2 (Equal variances) Independent samplesMatched pairs Experimental design? Unequal Equal

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 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).Xm14-01 –Can we conclude that ABS is effective? Identifying the appropriate technique

7 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? Identifying the appropriate technique

8 Question 1: Compare the accident rates Solution – continued Problem objective? Describing a single populationCompare two populations Data type? Interval Nominal Z test & estimator of p 1 -p 2 Z test & estimator of p 1 -p 2 A car had an accident: Yes / No

9 Solution – continued –p 1 = proportion of cars without ABS involved with an accident p 2 = proportion of cars with ABS involved with an accident –The hypotheses test H 0 : p 1 – p 2 = 0 H 1 : p 1 – p 2 > 0 Use case 1 test statistic Question 1: Compare the accident rates

10 Solution – continued –Use Test Statistics workbook: z-Test_2 Proportions(Case 1) worksheet 42   500 Do not reject H 0. Question 1: Compare the accident rates

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

12 Equal Solution - continued Population variances equal? Independent samplesMatched pairs Unequal Experimental design? Central location t- test & estimator of  1 -  2 (Equal variances) t- test & estimator of  1 -  2 (Equal variances) Run the F test for the ratio of two variances. Equal Question 2: Compare the mean repair costs per accident

13 Solution – continued –  1 = mean cost of repairing cars without ABS  2 = mean cost of repairing cars with ABS –The hypotheses tested H 0 :  1 –  2 = 0 H 1 :  1 –  2 > 0 –For the equal variance case we use Question 2: Compare the mean repair costs per accident

14 Solution – continued –To determine whether the population variances differ we apply the F test –From Excel Data Analysis we have (Xm14-01)Xm14-01 Do not reject H 0. There is insufficient evidence to conclude that the two variances are unequal. Question 2: Compare the mean repair costs per accident

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

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.