Chapter 11.docx

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Department
Quantitative Methods
Course
QMS 202
Professor
Bob Hudyma
Semester
Winter

Description
Chapter 11: Fundamentals of Hypothesis Testing: One-Sample Tests 11.1: Fundamentals of Hypothesis Testing Methodology The Null and Alternative Hypotheses - Null hypothesis (H ) - status quo; current belief in situation 0 o in terms of pop o if not rejected → failed to prove alt hypothesis o refer to specified val of pop parameter (μ) o equal sign regarding val of pop parameter: H :μ = 368 g 0 - Alternative Hypothesis (H ) 1 opposite of null hypothesis o whenever null hypothesis is false, alt is true o rep conclu/ research claim reached by rejected null hypothesis when sample info show null hypothesis is false o stat proof correct vs. null wrong o never equal sign regarding val of pop parameter: H :1≠ 368 g The Critical Value of the Test Statistic - hypothesis testing: how likely null hypothesis true based on info f/ sample - even if null hypothesis true, sample mean likely differ f/ val of parameter μ b/c variation f/ sampling - if sample stat close to pop parameter = insufficient evidence to reject null hypothesis - large diff b/ stat and hypothesized val of pop parameter = null hypothesis false - provides clear definitions for eval diff; quantify d/m/ process by computing prob of getting sample result if null hypothesis true 1. calc sample mean 2. test statistic - sample result = normal distribution, t distribution Regions of Rejections and Non-rejection - region of rejection - critical region = reject null hypothesis o val unlikely to occur if null hypothesis true o size dir related to risks involved using only sample evidence to d/ about pop parameter - region of non-rejection - not reject null hypothesis - critical value - divides non-rejection region f/ rejection region o depends on size of rejection of region Risks in Decision Making Using Hypothesis Testing - Type 1 Error - reject null hypothesis H 0hen it's true and shouldn't be rejected o Prob of Type 1 error occurring: α - Type 2 Error - don't reject null hypothesis H0when it's false and should be rejected = missed opp o Prob of Type 2 error occurring: β - Level of Significance (α) - prob of type 1 error happening o since you specify level of sig b4 hypothesis test is performed, risk is dir under your control o risk level depends on cost of making type 1 error o specify val for α → determine critical val that divide rejection and non-rejection regions - Confidence Coefficient (1 - α) - prob not reject null hypothesis H 0hen true and shouldn't be rejected o confidence level = (1 - α) x 100% - β Risk - prob of committing type 2 error o depends on diff b/ hypothesized and actual val of pop parameter o if diff large → β is small - Power of a Statistical Test (1 - β) - prob you will reject the null hypothesis when it's false and should be rejected - Risks in decision making: A delicate balance Actual Situation Statistical H 0rue H 0alse Decision Do not reject H0 Correct decision Type 2 Error Confidence = (1 - α) P (Type 2 error) = β Reject H Type 1 Error Correct decision 0 P (Type 1 error) = α Power = (1 - β) o reduce prob of making type 2 error →increase sample size = detect small diff → decreases β = increase prob of null hypothesis being false o control risk of type 1 error by reducing α o but mean doesn't change 11.2: Z Test of Hypothesis for the Mean (σ Known) Z Test for the Mean - when std dev σ is known and pop is normally distributed or if not normally distrib (sample large ebough for Central Limit Theorem take effect) - measures diff b/ observed sample mean and hypothesized mean μ The Critical Value Approach to Hypothesis Testing - Z STATcompared to critical val (expressed in std Z val = std error units) - decision rule: reject 0 if STAT> +1.96; or if STAT< -1.96 - using # - Z = +1.50 → between -1.96 and +1.96 = not reject H STAT 0 - take into account Type 2 Error say → there is insufficient evidence mean is diff f/ 368 g 6 Steps of Hypothesis Testing 1. state null hypothesis H0and alt hypothesis H 1 2. choose level of sig (α = based on relative importance of risks of committing Type 1 and 2 errors in prob) and sample size (n) 3. determine appropriate test stat and sampling distrib 4. determine critical val that divide rejection and non-rejection regions 5. collect sample data and compute val of test stat 6. make stat d/ and stat managerial conclu o test stat in non-rejection region = not reject null hypothesis o test stat in rejection region = reject null hypothesis The p-Value Approach to Hypothesis Testing - p-value/ observed level of significance - prob of getting test stat equal to or more extreme than sample result given null hypothesis H is true 0 - decision rule for rejecting: o p-val greater than or equal to α = don't reject null hypothesis o p-val less than α = reject null hypothesis - p-val for 2 tailed test: o find prob of getting test statSTATequal to or more extreme than ±1.50 std error units f/ centre of std normal distrib CASIO: STAT → F3(test) → F1(Z) → F1(1-S) Results: 1-Sample ZTest: 1-Sample or F6(DRAW) = normal distrib Data: F2(Var) μ ≠ 368 μ: F1(≠μ0) z = 1.5 μ0: 368 p = 0.133 σ: 15 = 372.5 : 372.5 n = 25 n: 25 Save Res: None EXE → Don't reject null hypothesis Find z critical val: STAT → F5(dist) → F1(Norm) → F3(InvN) Inverse Normal Results: Inverse Normal Data: F2(Var) X1Inv = -1.959964 Tail: F3(CNTR) X2Inv = 1.95996398 Area: 1-0.05 σ: 1 μ: 0 Save Res: None EXE - 5 Step p-Value Approach to Hypothesis Testing 1. state null hypothesis H0and alt hypothesis H 1 2. choose level of sig (α= based on relative importance of risks of committing type 1 and 2 errors in prob) and sample size (n) 3. determine appropriate test stat and sampling distrib 4. collect sample dat, compute val of test stat and compute p-val 5. make stat d/ and state managerial conclu o p-val greater than or equal to α = not reject null hypothesis o p-val less than α = reject null hypothesis o if p-val low,0H must go* A Connection Between Confidence Interval Estimation and Hypothesis Testing - confidence interval estimation o Ch 10: estim parameters - hypothesis testing o Ch 11: m/d/ about specified val of pop parameters; prove parameter more than, less than, not equal to specified val 11.3: t Test of Hypothesis for the Mean (σ Unknown) → Use Std Dev, S if pop not normally distrib, you can still use the t test if sample size is large (Central Limit Theorem) t Test for the mean - pop normally distrib, sample distrib of mean follows t distrib w/ n - 1 degrees of freedom - - hypothesis test prove mean increasing/decreasing The Critical Value Approach - 6 Steps of Hypothesis Testing 1. state null hypothesis 0 and alt hypothesis 1 2. choose level of sig (α = based on relative importance of risks of committing Type 1 and 2 errors in prob) and sample size (n) 3. determine appropriate test stat and sampling dis
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