chap11 - Business Statistics (BUSA 3101) Business Dr. Lari...

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Unformatted text preview: Business Statistics (BUSA 3101) Business Dr. Lari H. Arjomand lariarjomand@clayton.edu Chapter 11 Hypothesis Testing About the Differences Between Two Population Means H0 : µ 1 - µ2 = 0 Ha : µ 1 - µ2 # 0 Slide 2 Hypothesis Testing About the Differences Between Two Population Means Let µ1 equal the mean of population 1 and µ2 equal the mean of population 2. s The difference between the two population means is µ1 ­ µ2. s s Let n1 be the sample size of population 1 and n2 the sample size of population 2. x2 x1 Let equal the mean of sample 1 and equal the mean of sample 2. s Slide 3 Hypothesis Tests About µ 1 − µ 2: σ 1 and σ 2 Known Hypotheses H 0 : µ 1 − µ 2 ≥ D0 H 0 : µ 1 − µ 2 ≤ D0 H a : µ 1 − µ 2 < D0 H a : µ 1 − µ 2 > D0 Left­tailed Right­tailed H 0 : µ 1 − µ 2 = D0 H a : µ 1 − µ 2 ≠ D0 Two­tailed Test Statistic (Actual z Value) z= ( x1 − x2 ) − D0 2 σ12 σ 2 + n1 n2 Slide 4 Two cities, Bradford and Kane are separated only by the Conewango River. There is competition between the two cities. The local paper recently reported that with a standard deviation the mean household income of $7,000 for a sample of in Bradford is $38,000 with 35 households. At the 0.01 a standard deviation of significance level can we $6,000 for a sample of 40 conclude the mean income households. The same in Bradford is more? article reported the mean income in Kane is $35,000 Challenging Example 1 Slide 5 Step 4 State the decision rule. The null hypothesis is rejected if actual z is greater than critical z of 2.33 or if p ≤ .01 Step 1 State the null and alternate hypotheses. H0: µB < µK H1: µB > µK Step 3 Find the appropriate test statistic. Because both samples are more than 30, we can use z as the test statistic. Step 2 State the level of significance. The 0.01 significance level is stated in the problem. Solution Slide 6 Step 5: Compute the value of actual z and make a decision z= $38,000 − $35,000 ($6,000) 2 ($7,000) 2 + 40 35 = 1.98 Actual Z Because the actual Z = 1.98 < critical Z = 2.33, and since the p­value = 0.0239 > α = 0.01 the decision is to accept the null hypothesis. Thus we cannot conclude that the mean household income in Bradford is larger then the mean household income in Kane. H0: µB < µK H1: µB > µK Slide 7 Hypothesis Tests About µ 1 − µ 2: σ 1 and σ 2 Unknown and Small Sample When σ 1 and σ 2 are unknown, we will: • use the sample standard deviations s1 and s2 as estimates of σ 1 and σ 2 , and • use t table instead of Z table. •Assuming at least one of the sample is n < 30 Slide 8 Hypothesis Tests About µ 1 − µ 2: σ 1 and σ 2 Unknown and Small Sample s Hypotheses H 0 : µ 1 − µ 2 ≥ D0 H 0 : µ 1 − µ 2 ≤ D0 H a : µ 1 − µ 2 < D0 H a : µ 1 − µ 2 > D0 Left­tailed s Right­tailed H 0 : µ 1 − µ 2 = D0 H a : µ 1 − µ 2 ≠ D0 Two­tailed Test Statistic (Actual t)—When n < 30 and σ is unknown: t= ( x1 − x2 ) − D0 2 s12 s2 + n1 n2 Slide 9 EXCEL APPLICATION When in a problem raw data are given, then you may use either Data Analysis Add-Ins or SWStat+ Add-Ins in Excel to solve the problem. See next slides. Hypothesis Tests About µ 1 = µ 2 When σ 1 and σ 2 Unknown & Small Samples Challenging Example #1 Using SWStat+ and Data Analysis in Excel s Ariana Corporation wants to increase the Ariana productivity of its line workers. To do so, two different programs have been suggested to help increase productivity. Twenty employees, making up a sample, have been randomly assigned to one of the two programs and their assigned output for a day's work has been recorded. The results are given on the next slide. results Slide 11 Example #1 Continued Program A Program B 150 150 130 120 120 135 180 160 145 110 185 175 220 150 190 120 180 130 175 220 Slide 12 Example #1 Continued Questions s s s a. State the null and alternative hypotheses. b. Use Excel, and at 95% confidence test to determine if the means of the two populations are equal, and c. Explain your answer Slide 13 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 Unknown & Small Samples Using Data Analysis in Excel Excel’s “t­Test: Two Sample Assuming Unequal Variances” Tool s Step 1 Select the Tools menu Step 2 Choose the Data Analysis option Step 3 Choose t­Test: Two Sample Assuming Unequal Variances from the list of Analysis Tools … continued Slide 14 Solution to #1 Using Data Analysis in Excel s Excel Dialog Box Slide 15 Solution to #1 Using Data Analysis in Excel H0: µ1 = µ2 H : µ # µ Slide 16 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 Unknown & Small Samples s SWStat+ “t­Test: Two Sample Assuming Unequal Variances” Tool Step 1 Create Data area Step 2 Choose Statistics, and then choose the Intervals and Tests option Step 3 From Type, Choose Two Samples, Different Means, Unequal Variances, t­Test … continued Slide 17 Solution to #1 Using SWStat+ (Creating Data Area) Data Area Slide 18 Solution to #1 Using SWStat+ (Selecting Intervals and Tests Option) Slide 19 Solution to #1 Using SWStat+ (Results) H0: µ1 = µ2 H : µ # µ Slide 20 Hypothesis Tests About µ 1 = µ 2 When σ 1 and σ 2 Unknown Challenging Example #2 s Information regarding the ACT scores of samples of twenty two students in two different majors at CSU is given in next slide. s Questions: (a) Use Excel, and at 95% confidence test to determine whether there is a significant difference in the means of the two populations, and (b) explain your answer. Slide 21 DATA Hypothesis Test H0: µ1 = µ2 Ha: µ1 # µ2 Slide 22 Solution to #2 Using Data Analysis in Excel Slide 23 Solution to #2 Using SWStat+ (Results) Two Samples, Different Means, Unequal Variances, t Test H0: µ1 = µ2 Ha: µ1 # µ2 Slide 24 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 are Known s s s If samples are large (n1≥30 and n2≥30) and also if population standard deviations (σ1 and σ2) are given then you can use either Data Analysis or SWStat+ in Excel to solve the problem. Note that in this case, the option that you use in Data Analysis is z­Test: Two Sample for Means and the option that you use in SWStat+ is Two Sample, diff Variances/SDs: F See next example. Slide 25 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 are Known s Example: Par, Inc. Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” In a test of driving distance using a mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics appear on the next slide. Slide 26 Hypothesis Testing of µ1 ­ µ2: σ 1 and σ 2 Known s Example: Par, Inc. Sample Size Sample Mean Sample #1 Sample #2 Par, Inc. Rap, Ltd. 120 balls 80 balls 275 yards 258 yards Given that the two population standard deviations are known as: σ 1 = 15 yards and σ 2 = 20 yards, then known Can we conclude, using α = 0.01, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls? Slide 27 Hypothesis Testing of µ1 ­ µ2 When σ 1 and σ 2 are Known p –Value and Critical Value Approaches The hypotheses is: H0: µ1 ­ µ2 < 0 Ha: µ1 ­ µ2 > 0 where: µ1 = mean distance for the population of Par, Inc. golf balls µ2 = mean distance for the population of Rap, Ltd. golf balls The level of significance is: α = .01 Slide 28 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 are Known s Excel’s “z­Test: Two Sample for Means” Tool Step 1 After data entry, then select the Tools menu Step 2 Choose the Data Analysis option Step 3 Choose z­Test: Two Sample for Means from the list of Analysis Tools … continued Slide 29 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 are Known s Excel Dialog Box H0: µ1 ­ µ2 < 0 H : µ ­ µ > 0 OR H0: µ1 ≤ µ2 H : µ > µ Slide 30 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 are Known s 1 2 3 4 5 6 7 8 9 10 11 12 13 Excel Value Worksheet A Par 255 270 294 245 300 262 281 257 268 295 249 291 BC D Rap 266 z-Test: Two Sample for Means 238 243 277 Mean 275 Known Variance 244 Observations 239 Hypothesized Mean Difference 242 z 280 P(Z<=z) one-tail 261 z Critical one-tail 276 P(Z<=z) two-tail 241 z Critical two-tail Note: Rows 14­121 are not shown. E F P ar, Inc. Rap, Ltd. 235 218 225 400 120 80 0 6.483545607 4.47959E-11 2.326347874 8.95919E-11 2.575829304 Slide 31 Conclusion s s Because p–value = 0 < α = 0.01 and also because actual z = 6.49 > critical z = 2.33 we reject H0. In other words: At the 0.01 level of significance, the sample evidence indicates that the mean driving distance of Par, Inc. golf balls ( µ1) is greater than the mean driving distance of Rap, Ltd. golf balls (µ2). In other words H0: µ1 ­ µ2 < 0 Ha: µ1 ­ µ2 > 0 OR H0: µ1 ≤ µ2 Ha: µ1 > µ2 Slide 32 End of Chapter 11 DANCER ...
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