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Lecture

# Lecture 5

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Ryerson University

Quantitative Methods

QMS 202

Clare Chua

Winter

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272011 Hypothesis Test Hypothesis Test Two Populations Two-Sample Tests One Population 2 MEANS 2 Proportions Chapter 11 Page 452 Sample Independent Sample Check the 4 dependent Objectives conditions Normal? No 1. Comparing The Means of Two Independent STOP If yes Yess Populations Paired t-test No Z-test Z-test for the difference between two means. No 2 2 81and 8 2nown? Equal 8 ? unequal 8 Pooled-Variance t-test for the difference between Two Means. Yes Yes 2. Comparing The Means of Two Related Populations T-test not Pooled variance 3. Comparing Two Population Proportions Z-test Pooled Variance t-test Z-Test 4. F-Test For The Difference Between Two Variances Comparing Dependent and Independent There are two types of dependent samples: Samples Characterized by a Measurement, Characterized by matching or followed by an intervention of pairing of the observations. oZ k some type, and then another measurement Example: ZB Suppose you are selling a house Called a before and after study. and you asked for two IndepeendenttSampples DepeendenttSammplee Example: The samples chosen at random are Dependent samples are Suppose we want to show that, by appraisals of your property. nott eated to each other. characterized by a measurement, placing speakers in the productionSuppose an industrial then some type of intervention, psychologist wishes to study We wish to study the mean incomes followed byy another measurement. area and playing soothing music, offommpannessX and Y. wee ae able o ncceasee the intelectuallsmiiarries off We select a random sample of 28 Paired samples are also production. newly married couples. She employees from the Company X and dependent because the same We begin by selecting a sample of selects a sample of a sample of 19 employees in individual or item is a member of workers and measuring their newlyweds. Company Y. both samples. output under current conditions, The speakers are then installed inNext she administers a standard A person cannot be an employee in Examples: 10 participants in a intelligence test to both the both companies. marathon were weighed prior to the production area, and we again man and woman to determine The samples are indeppenddentt, and after competing in the race. measure the output of the same that is, unrelated. We wish to study the mean workers. the difference in the scores amount of weight loss. There are 2 measurements: (1) Notice the matching that before placing the speakers occurred: comparing the (2) After placing the speakers. scores of the man and the woman. Comparing Dependent and Independent The seven-step Method of Hypothesis Testing Section 11.1 Samples Section 11.2 Steps: IndeeppenndeennttSaamp pless DeepenndenntSaampplee Paired t-test: 1. Define the parameters: 2 1nd 2 2 z-test: Test of two indep Test of two dep pop, 8 equal but 2. State the hypothesis pop, 8 Known Ho: 2 = 2 (or Ho: 2 - 2 = 0) unknown t D D 1 2 1 2 z (1 x2) (1 2) calc SD Ha: 21 2 2 (or Ha: 1 - 2 0) calc 2 2 n 3. Determine the appropriate test (refer to the flowchart) 1 2 Where D is difference scores 4. State the conditionsassumptions: (1) Normal? (2) Equal variance? n1 n2 dffn-1 5. Determiine hee crrcallvalue haatdivide the ejeccton anddnoon-rejecton region 6. Calculate the test statistics value t-test: Test of two indep pop, 7. Make the statistical decision and Managerial conclusion t-test: Test of twoep pop, 8 unequal and unknown 8 equal but unknown To reject Ho or DO NOT reject Ho t (x 1 x 2 2 1 2 2) Compare the test statistics with the critical value (critical value approach) (x 1 x 2 (1 2) calc 2 2 or tcalc s1 s2 2 1 1 n1 n2 compare p-value with . (p-value approach) s p n n ) s2 s2 1 2 ( 1 ) ( 2 ) df=(1 -1)+(2 -1)1n 2n -2 n 1 n 2 2 (n 1 1)s1 (n2 1)s2 df 4 4 sp s1 s2 (n1 1)(n 2 1) n 1n 1 1) n 2n 2 1) 1 www.notesolution.com 272011 Illustration: Calculator steps Z Test for Difference Between Two Means (8 known) A company that makes bolts that are used on an automotive From the Main Menu select: component uses two machines to make these bolts. It has been STAT F3(test) F1(Z) F2(2-S)then enter the following items: diameters made by machine 1 is 0.025 mm. and the standard the bolt 2-Sample ZTest deviation of the bolt diameters of machine 2 is 0.022 mm. Both Data : F2(Var)z machines have a dial to set for the desired diameter. Recently they 1 : F2(<2)z used both machines to fill a large order. The customer found that 1 : 0.025 EXE maanyoffhebols om a ceran pacage were oo age andmade a 2 : 0.022 EXE complaint. It was determined that the package in question was 1 : 5.023 EXE the bolts from both machines to test to see whether the meanof n1 : 100 EXE diameter of the bolts from machine 2 was significantly larger than 2 : 5.031 EXE the mean diameter from machine 1 when the dial was set to the same diameter on each machine. The sample of 100 bolts from n2 : 100 EXE machine 1 had a mean diameter of 5.023 mm and a sample of 100 dial on both machines was set at 5.00 mm. At the 5% level ofe Now key EXE or F1(Calc) significance what is the conclusion? CFX-9850GB Calculator The calculator will now show the Z Test for Difference Between Two Means (8 known) results: A company that makes bolts that are used on an automotive component uses two machines to make these bolts. It has been 2-sample ZTest determined by past studies that the standard deviation of the bolt diameters made by machine 1 is 0.025 mm. and the standard 1 < 2 deviation of the bolt diameters of machine 2 is 0.022 mm. Both machines have a dial to set for the desired diameter. Recently they z =-2.4022 maanyoffhebols om a ceran pacage were oo age andmade a found that complaint. It was determined that the package in question was p =88.114665EE-003 ((=00.00081144655) made by machine 2. The manufacturer decided to take samples of 1 =5.023 the bolts from both machines to test to see whether the mean diameter of the bolts from machine 2 was significantly larger 2 =5.031 than the mean diameter from machine 1 when the dial was set to machine 1 had a mean diameter of 5.023 mm and a sample of 100om n1 =100 bolts from machine 2 had a mean diameter of 5.031 mm when the dial on both machines was set at 5.00 mm. At the 5% level of n2 =100 significance what is the conclusion? Illustration Z Test for Difference Between 8wknown)s ( The operations manager at a light bulb factory wants to determine if there is any difference in the average life expectancy of bulbs manufactured on two different types of machines. The process standard deviation of machine I is 110 hours and of machine II is 125 hours. Solution will be shown in class A random sample of 25 light bulbs obtained from maachine IIndcates a samppe meean off375 hours,andd a similar sample of 25 from machine II indicates a sample mean of 362 hours. a) Using the 0.05 level of significance, is there any evidence of a difference in the average life of bulbs produced by the two types of machines? b) Compute p-value in (a) and interpret its meaning. 2 www.notesolution.com

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