Introduction I Introduction I Introduction II Introduction II - PowerPoint PPT Presentation

Outline: Chapter 8 Hypothesis Testing Outline: Chapter 8 Hypothesis Testing 2 � Introduction I d i � Hypothesis Testing 204312 PROBABILITY AND 204312 PROBABILITY AND � Significance Testing (8.1 Y&G) RANDOM PROCESSES FOR � Binary Hypothesis Testing (8 2 Y&G) � Binary Hypothesis Testing (8.2 Y&G) COMPUTER ENGINEERS COMPUTER ENGINEERS � Multiple Hypothesis Test (8.3 Y&G) Lecture 11: � Significance Levels Chapter 8 p � Z-Tests 1st Semester, 2007 � Summary Monchai Sopitkamon, Ph.D. Introduction I Introduction I Introduction II Introduction II � Statistical inference – analyze observations of an � Statistical inference consists of 3 steps: experiment to derive a conclusion based on Perform an experiment, f properties of RVs. 1. Observe an outcome, and 2. � Last chapter presented two types of statistical p p yp State a conclusion based on prob theory 3. inference for model parameters: point estimation Goal of making conclusion: achieve the highest g g and confidence interval estimation. and confidence interval estimation. � possible accuracy � This chapter presents two more types of inference: significance testing and hypothesis testing. significance testing and hypothesis testing

Hypothesis Testing : Hypotheses I Hypothesis Testing : Hypotheses II (8 2 1) (8.2.1) (8.2.1) � Allows assessing of the plausibility or credibility of � A t wo-sided set of hypothesis is a specific statement or hypothesis. versus H A : µ ≠ µ 0 H 0 : µ = µ 0 � Hypothesis Tests of a Population Mean For a specified value of µ 0 , and a one-sided set of hypothesis is either A null hypothesis H 0 for a population mean µ is a A null hypothesis H 0 for a population mean µ is a H 0 : µ ≤ µ 0 versus H A : µ > µ 0 statement that designates possible values for the population mean. p p or or The alternative hypothesis H A is the “opposite” of the H 0 : µ ≥ µ 0 versus H A : µ < µ 0 null hypothesis. yp Hypothesis Testing : Hypotheses III Hypothesis Testing: Hypotheses IV (8.2.1) (8.2.1) � Ex.14 pg.342: Metal Cylinder Production � Ex.48 pg.343: Car Fuel Efficiency The two-sided hypotheses are The one-sided hypotheses are H 0 : µ = 50 versus H A : µ ≠ 50 H 0 : µ ≥ 35 versus H A : µ < 35 The null hypothesis states that the machine produces The null hypothesis states that the manufacturer’s claim cylinders with the mean diameter of 50 mm, and the cylinders with the mean diameter of 50 mm, and the regarding the fuel efficiency of its cars is correct, and regarding the fuel efficiency of its cars is correct, and alternative hypothesis states that the mean diameter is the alternative hypothesis states that the opposite. not equal to 50 mm. q

Hypothesis Testing: Interpretation of p - Hypothesis Testing: Interpretation of p - Values (8.2.2) Values II (8.2.2) � p -Values A data set can be used to measure the plausibility of a null hypothesis H 0 through the construction of a p -value. The smaller the p -value, the less plausible is the null hypothesis. The p -value can also be called level of significance . P -value construction Hypothesis Testing: Interpretation of p - Hypothesis Testing: Interpretation of p - Values III (8.2.2) Values IV (8.2.2) P -value interpretation P -value interpretation P -value interpretation P -value interpretation � Rejection of the Null Hypothesis � Rejection of the Null Hypothesis � Acceptance of the Null Hypothesis A p- value smaller than 0.01 indicates that the null A p- value larger than 0.1 indicates that the null h hypothesis H 0 is not a plausible statement. th i H i t l ibl t t t hypothesis H 0 is a plausible statement. The null hypothesis H 0 is then rejected in favor of The null hypothesis H 0 is then accepted . However, 0 the alternative hypothesis H A . th lt ti h th i H H 0 has not been proven to be true either.

Hypothesis Testing: Interpretation of p - Hypothesis Testing : Interpretation of p - Values V (8.2.2) Values VI (8.2.2) P -value interpretation P -value interpretation � Ex.14 pg.342: Metal Cylinder Production The two-sided hypotheses are H 0 : µ = 50 versus H A : µ ≠ 50 � Intermediate p -Values With a small p-value, the null hypothesis is rejected and the machine is shown to be A p- value ranges 1% - 10% indicates that data analysis is inconclusive. miscalibrated. There is some evidence that the null hypothesis is not With a large p -value, the null hypothesis is plausible, but the evidence is not overwhelming. accepted and it can be concluded that there is A larger data set or a cutoff value of 0 05 may be A larger data set or a cutoff value of 0.05 may be no evidence that the machine is calibrated used to provide better conclusion. incorrectly. Hypothesis Testing: Calculation of p - Hypothesis Testing: Interpretation of p - Values VII (8.2.2) Values I (8.2.3) � Two-sided t -Test T id d t T � Ex.48 pg.343: Car Fuel Efficiency The p -value for the two-sided hypothesis testing problem bl The one-sided hypotheses are versus H A : µ ≠ µ 0 H 0 : µ = µ 0 H 0 : µ ≥ 35 versus H A : µ < 35 b based on a data set of n observations w/ a d d t t f b ti / sample mean and a sample SD s is x If a small p-value is obtained, the null hypothesis is p value = 2 x P ( X ≥ | t |) p -value = 2 x P ( X ≥ | t |) rejected and the manufacturer’s claim is where the RV X has a t -distribution w/ n – 1 incorrect. degrees of freedom, and degrees of freedom, and A large p -value indicates that there is not enough ( ) − μ n x = 0 t evidence to conclude that the manufacturer’s s which is known as the t -statistic . hi h i k th t t ti ti claim is correct.

Hypothesis Testing: Calculation of p - Hypothesis Testing: Calculation of p - Values II (8.2.3) Values III (8.2.3) � Ex Consider the hypothesis � Ex. Consider the hypothesis P -value for tw o-sided t -test H 0 : µ = 10.0 versus H A : µ ≠ 10.0 suppose that a data set has n = 15, = x 10.6, and s = 1.61. The t -statistic is ( ( ) ) ( ( ) ) − μ − 15 15 10 10 . 6 6 10 10 . 0 0 n x = = = 0 1 . 44 t 1 . 61 s The p -value = 2 x P ( X ≥ 1.44) where the RV X has a t -distribution w/ n – 1 = 14 degrees of freedom g p -value = 2 x 0.086 = 0.172 Since p -value = 0 172 > 0 1 then the null Since p value 0.172 > 0.1, then the null hypothesis that µ = 10.0 should be accepted. Excel sheet Hypothesis Testing: Calculation of p - Hypothesis Testing: Calculation of p - Values IV (8.2.3) Values V (8.2.3) � One-sided t -Test O Based on a data set of n observations w/ a sample mean and a sample SD s, the p -value x for the one-sided hypothesis testing problem H 0 : µ ≤ µ 0 versus H A : µ > µ 0 is p -value = P ( X ≥ t ) and p -value for the one-sided hypothesis testing problem H 0 : µ ≥ µ 0 versus H A : µ < µ 0 is p -value = P ( X ≤ t ) where the RV X has a t -distribution w/ n – 1 ( ( ) ) d degrees of freedom, and f f d d − μ n x = 0 t s Tw o-sided p -value calculation

Hypothesis Testing: Calculation of p - Hypothesis Testing: Calculation of p - Values VI (8.2.3) Values VII (8.2.3) � Ex. Consider the hypothesis E C id th h th i H 0 : µ ≤ 125.0 VS H A : µ > 125.0 Suppose that a sample mean = 122.3 is x observed. What’s the p -value ? p Since the sample mean corresponds to x population mean µ contained within the null population mean µ contained within the null hypothesis, the p -value must be > 0.5, and therefore, the null hypothesis that µ ≤ 125.0 therefore the null hypothesis that µ ≤ 125 0 should be accepted. P -value calculation for a P -value larger than 0 .5 0 one-sided problem id d bl for a one-sided t -test f Hypothesis Testing: Calculation of p - Hypothesis Testing: Calculation of p - Values VIII (8.2.3) Values VIII (8.2.3) � Suppose instead that a sample mean = � Suppose instead that a sample mean = x x 128.4 is observed, with n = 20 and s = 16.9. Since = 128.4 > µ 0 = 125, this suggest S 128 4 125 h x that the null hypothesis H 0 is false. How plausible is H 0 ? ? ( ) ( ) − μ − 20 128 . 4 125 n x The t -statistic is = = = 0 0 . 9 t 16 16 . 9 9 s s The p -value = P ( X ≥ 0.9) = 0.19 Since p value = 0 19 > 0 1 the null Since p -value = 0.19 > 0.1, the null P -value larger than 0 .5 0 for a one- hypothesis is accepted. sided t -test Excel sheet