MTH120_Chapter11

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MTH120_Chapter11

  1. 1. 11- 1 Chapter Eleven Two Sample Tests of HypothesisGOALSWhen you have completed this chapter, you will be able to: ONE Understand the difference between dependent and independent samples.TWOConduct a test of hypothesis about the difference between two independentpopulation means when both samples have 30 or more observations.THREEConduct a test of hypothesis about the difference between two independentpopulation means when at least one sample has less than 30 observations.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  2. 2. 11- 2 Chapter Eleven continued Two Sample Tests of HypothesisGOALSWhen you have completed this chapter, you will be able to:FOURConduct a test of hypothesis about the mean difference between paired ordependent observations.FIVEConduct a test of hypothesis regarding the difference in two populationproportions.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  3. 3. 11- 3 Comparing two populations We wish to know whether the distribution of the differences in sample means has a mean of 0. If both samples contain at least 30 observations we use the z distribution as the test statistic.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  4. 4. 11- 4 Comparing two populations No assumptions about the shape of the populations are required. The samples are from independent populations. The formula for computing the value of z is: X1 − X 2 z= 2 2 s1 s 2 + n1 n2McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  5. 5. 11- 5 EXAMPLE 1 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 the mean household income in Bradford is $38,000 with a standard deviation of $6,000 for a sample of 40 households. The same article reported the mean income in Kane is $35,000 with a standard deviation of $7,000 for a sample of 35 households. At the .01 significance level can we conclude the mean income in Bradford is more?McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  6. 6. 11- 6 EXAMPLE 1 continued  Step 1: State the null and alternate hypotheses. H0: µB µK ; H1: µB > µKStep 2: State the level of significance. The .01significance level is stated in the problem.Step 3: Find the appropriate test statistic. Because bothsamples are more than 30, we can use z as the test statistic.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  7. 7. 11- 7 Example 1 continued  Step 4: State the decision rule. The null hypothesis is rejected if z is greater than 2.33.Step 5: Compute the value of z and make a decision. $38,000 − $35,000 z= = 1.98 ($6,000) 2 ($7,000) 2 + 40 35McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  8. 8. 11- 8 Example 1 continued The decision is to not reject the null hypothesis. We cannot conclude that the mean household income in Bradford is larger. The p-value is: P(z > 1.98) = .5000 - .4761 = .0239McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  9. 9. 11- 9 Small Sample Tests of Means  The t distribution is used as the test statistic if one or more of the samples have less than 30 observations. The required assumptions are: 1. Both populations must follow the normal distribution. 2. The populations must have equal standard deviations. 3. The samples are from independent populations.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  10. 10. 11- 10 Small sample test of means continued  Finding the value of the test statistic requires two steps. 1. Pool the sample standard deviations. 2 2 (n1 − 1) s1 + (n2 − 1) s 2 s2 = p n1 + n2 − 2 Determine the value of t from the following formula. X1 − X 2 t= 2 1 1  sp  +    n1 n2 McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  11. 11. 11- 11 EXAMPLE 2 A recent EPA study compared the highway fuel economy of domestic and imported passenger cars. A sample of 15 domestic cars revealed a mean of 33.7 mpg with a standard deviation of 2.4 mpg. A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of 3.9. At the .05 significance level can the EPA conclude that the mpg is higher on the imported cars?McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  12. 12. 11- 12 Example 2 continuedStep 1: State the null and alternate hypotheses. H0: µD µI ; H1: µD < µI Step 2: State the level of significance.The .05 significance level is stated in the problem. Step 3: Find the appropriate test statistic. Both samples are less than 30, so we use the t distribution.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  13. 13. 11- 13 EXAMPLE 2 continued Step 4: The decision rule is to reject H0 if t<-1.708. There are 25 degrees of freedom. Step 5: We compute the pooled variance: 2 2 (n1 − 1)( s1 ) + (n 2 − 1)( s 2 ) s2 = p n1 + n 2 − 2 (15 − 1)(2.4) 2 + (12 − 1)(3.9) 2 = = 9.918 15 + 12 − 2McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  14. 14. 11- 14 Example 2 continued We compute the value of t as follows. X 1 −X 2 t = 1 1  s2  p +   n1 n2   33.7 −35.7 = =− .640 1 1 1  8.312 +  15 12  H0 is not rejected. There is insufficient sample evidence to claim a higher mpg on the imported cars.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  15. 15. Hypothesis Testing Involving Paired 11- 15 Observations  Independent samples are samples that are not related in any way.Dependent samples are samples that are paired orrelated in some fashion. For example: If you wished to buy a car you would look at the same car at two (or more) different dealerships and compare the prices. If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  16. 16. Hypothesis Testing Involving Paired 11- 16 Observations Use the following test when the samples are dependent: d t= sd / n  where d is the mean of the differences  s is the standard deviation of the differences d  n is the number of pairs (differences)McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  17. 17. 11- 17 EXAMPLE 3  An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis. A random sample of eight cities revealed the following information. At the .05 significance level can the testing agency conclude that there is a difference in the rental charged?McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  18. 18. 11- 18 EXAMPLE 3 continued City Hertz ($) Avis ($) Atlanta 42 40 Chicago 56 52 Cleveland 45 43 Denver 48 48 Honolulu 37 32 Kansas City 45 48 Miami 41 39 Seattle 46 50McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  19. 19. 11- 19 EXAMPLE 3 continued Step 1: H0 : µd = 0 H1: µd ≠ 0Step 2: H0 is rejected if t < -2.365 or t > 2.365. We use the t distribution with 7 degrees offreedom.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  20. 20. 11- 20 Example 3 continued City Hertz Avis d d2 Atlanta 42 40 2 4 Chicago 56 52 4 16 Cleveland 45 43 2 4 Denver 48 48 0 0 Honolulu 37 32 5 25 Kansas City 45 48 -3 9 Miami 41 39 2 4 Seattle 46 50 -4 16McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  21. 21. 11- 21 Example 3 continued Σd 8.0 d= = = 1.00 n 8 Σd 2 − ( Σd ) 2 78 − 82 sd = n = 8 = 3.1623 n −1 8−1 d 1.00 t= = = 0.894 sd n 3.1623 8McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  22. 22. 11- 22 Example 3 continued Step 3: Because 0.894 is less than the critical value, do not reject the null hypothesis. There is no difference in the mean amount charged by Hertz and Avis.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  23. 23. 11- 23 Two Sample Tests of Proportions  We investigate whether two samples came from populations with an equal proportion of successes.  The two samples are pooled using the following formula. X1 + X 2 pc = n1 + n2 where X1 and X2 refer to the number of successes in the respective samples of n1 and n2.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  24. 24. 11- 24 Two Sample Tests of Proportions continued The value of the test statistic is computed from the following formula. p1 − p 2 z= pc (1 − pc ) pc (1 − pc ) + n1 n2McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  25. 25. 11- 25 Example 4 Are unmarried workers more likely to be absent from work than married workers? A sample of 250 married workers showed 22 missed more than 5 days last year, while a sample of 300 unmarried workers showed 35 missed more than five days. Use a .05 significance level.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  26. 26. 11- 26 Example 4 continued The null and the alternate hypothesis are: H0: : U U M H1: : U > >M The null hypothesis is rejected if the computed value of z is greater than 1.65.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  27. 27. 11- 27 Example 4 continued The pooled proportion is 35 + 22 pc = 300 + 250 The value of the teat statistic is 35 22 − z= 300 250 = 1.10 .1036(1 −.1036) .1036(1 −.1036) + 300 250McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  28. 28. 11- 28 Example 4 continued The null hypothesis is not rejected. We cannot conclude that a higher proportion of unmarried workers miss more days in a year than the married workers.The p-value is: P(z > 1.10) = .5000 - .3643 = .1457McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

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