# Bài giảng Probability & Statistics - Lecture 9: Hypothesis testing - Bùi Dương Hải

Example Example 9.2. In the past, average productivity of worker is 120. To improve productivity, new management procedure is applied. Recently survey of 25 workers shows sample mean is 126.2, sample variance is 225.  Assumed that productivity is Normality,  (a) Test the hypothesis that average productivity has increased, at significant level of 5% and 1%  Sample: 9 = 25; 7̅ = 126.2; 8< = 225  Then 8 = 15

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Lecture 9. HYPOTHESIS TESTING  Statistical Hypothesis  Error Types  T-test for Mean  Z-test for Population  Chi-sq test for Variance  [1] Chapter 9 + 11  pp. 382 – 432; 488 - 491 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 1 9.1. Statistical Hypothesis  Hypothesis: statement about statistical issue  Testing: test for the “True” or “False” of hypothesis  Example:  Government reports that average income is \$2400  Expected value of price is \$10  Mean of consumption was 2 mil.VND PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 2 Hypotheses Pair  For parameter ∶Null hypothesis :Alternative hypothesis  There are 3 types  : ≤ : > one-sided, upper tail  : ≥ : < one-sided, lower tail  : = : ≠ two-sided, two tails PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 3 Example Ex. Testing the statement that “average income of habitant is \$2000” 0: = 2000 : the statement is True 1: ≠ 2000 : the statement is False Example 9.1. What is hypotheses pair for the following statements: (a) The average income of habitant is greater than 2000 (b) The proportion of male in customers is less than 50% (c) The variability of price is more than 20 USD2 (d) The dispersion of price is less than 20 USD PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 4 Decision & Errors  Decision: Reject or Accept H0  Two types of Errors Error type 1: Reject the True hypothesis Error type 2: Accept the False hypothesis  Allow a given Type 1 error probability, minimize Type 2 error probability PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 5 Decision H0 is True H0 is False Accept H0 Correct decision Type 2 error Reject H0 Type 1 error Correct decision Testing Procedure  Probability for Type 1 Error: Significant Level,  Calculate Critical value correspond to  Indicate Reject Area for H0  From sample: calculate Statistical value  Rule:  Statistical value is in the Reject Area: Reject H0  Stat. value is not in the Reject Area: Not Reject H0 (Not enough evident to reject H0) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 6 Testing Procedure  Step 1: Setting up Hypothesis pair, significant level  Step 2: Data gathering  Step 3: Calculating statistical value  Step 4: Comparing statistical value with critical value  Step 5: Concluding about the hypothesis  There is Probability value (P-value) of the test  − < :reject  − > : not reject PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 7 9.2. T-test for Mean  Testing population mean with a value  Significant level:  Hypotheses pairs : ≤ : > : ≥ : < : = : ≠  Only one Statistical value: = ̅ /  Critical value and Reject Area are different PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 8 T-test for Mean Statistical value Hypotheses pair Critical value Reject H0 = −̅ / : ≤ : > > : ≥ : < − < − : = : ≠ / ||> / PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 9 Example Example 9.2. In the past, average productivity of worker is 120. To improve productivity, new management procedure is applied. Recently survey of 25 workers shows sample mean is 126.2, sample variance is 225.  Assumed that productivity is Normality,  (a) Test the hypothesis that average productivity has increased, at significant level of 5% and 1%  Sample: = 25; =̅ 126.2; = 225  Then = 15 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 10 Example  Hyp. Pair: : ≤ 120 : > 120  Stat. value: = . / =  At 5%: Critical value: . =  At 1%: Critical value: . = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 11 P-value of the Test  There is one Probability value ∗ that Statistical value equal Critical value  Probability value of the test: −  The rule  − <  Reject  − >  Not Reject  Accurate − is calculated by software  Using table, find the interval of − PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 12 Example  Example 9.2 (cont.)  (b) At 5%, test the hypothesis that recent average productivity equals 130  (c) Which interval contains the − of the test in question (b): (0 – 1%); (1% – 5%); (5% – 10%); > 10%  (d) At 10%, test the hypothesis that recent average productivity is less than 135, and estimate the − of the test PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 13 9.3. Z-test for Proportion PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 14 Statistical value Hypotheses pair Critical value Reject H0 = −̅ 1 − / : ≤ : > > : ≥ : < − < − : = : ≠ / ||> / Example Example 9.3. Last year, the proportion of visitors buying goods is 20%. Recent year, in observed 200 visitors, 52 of them buy at least one item. (a) With significant level at 5%, testing the hypothesis that the buying proportion has increased (b) Estimate the − of the test (c) At significant level of 1%, test the claim that Proportion is 30%, and estimate the − PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 15 9.3. Chisq-test for Variance PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 16 Statistical value Hypotheses pair Reject H0 = ( − 1) : ≤ : > > : ≥ : < < : = : ≠ > / Or < / Example Example 9.4. The required limit for the variability of fruits’ weight is 4g2. For sample of 20 fruits, sample variance is 5g2. (a) With the significant level of 5%, test the claim that the variability of fruits’ weight exceeds the required limit, assumed that the it is normal distributed. (b) Test the hypothesis that Population Standard deviation is less than 4g PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 17 Key Concepts  Hypotheses pair  Types of Error  Significant level  Critical value & Statistical value  Reject and Not reject H0  − of the test  T-test, Z-test, Chisq-test PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 18 Exercise [1] Chapter 9, 11  (387) 1, 2, (389) 5, 7  (402) 10, 15, 17  (408) 23, 25, 27, 29, 34  (414) 36, 38, 40, 43  (491) 4, 11  Case Problem 1 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 19