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Statistical Inference: A Review of Chapters 12 and 13. Chapter 14. 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.
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Statistical Inference:A Review of Chapters 12 and 13 Chapter 14
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.
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
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
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.
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?
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?
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
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
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
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
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
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
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)
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.
Checking required conditions • The two populations should be normal (or at least not extremely nonnormal)
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.