Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 1. Explain the the difference between dependent and independent samples. 2. Conduct a test of hypothesis and obtain a confidence interval estimate for the difference between … When you have completed this chapter, you will be able to: … two population means using matched pair sample. … two population proportions. … two population means using independent samples.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Step 5 Hypothesis Testing Do not reject H 0 Reject H 0 and accept H 1 Compute the value of the test statistic and make a decision

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. If both samples contain at least 30 observations we use the z distribution as the test statistic. We wish to know whether the the distribution of the differences in sample means has a mean of 0.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The samples are from independent populations. The formula for computing the value of z is: No assumptions about the of the populations are required. z n2n2 s n1n1 s   1 X  2 X

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Two colleges are located in York Region. The local paper recently reported that the mean starting salary of a graduate from College 1 is $38,000 with a standard deviation of $6,000 for a sample of 40 graduates. The same article reported the mean starting salary of a graduate from College 2 is $35,000 with a standard deviation of $7,000 for a sample of 35 graduates. At the.01 significance level can we conclude the mean salary of College 1 is more?

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5 H 0 : µ 1 = µ 2 H 1 : µ 1 > µ 2  = 0.01 Because both samples are more than 30, the test statistic is Z Reject H 0 if z > 2.33 Do not reject the null hypothesis; insufficient evidence n s n s XX z    = 1.98 $38,000 - $35, )000,7($ ),6($ 22   z

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. State the decision rule Step 4 The p-value is: p(z > 1.98) p(z > 1.98) = = Look up in Table Since p of.0239 is greater than the chosen alpha of 0.01, there is insufficient evidence to reject H

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Small Sample Tests of Means If one or more of the samples contain less than 30 observations we use the t distribution as the test statistic. Required Assumptions: Both populations must follow the normal distribution. The populations must have equal standard deviations. The samples are from independent populations.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.  Pool the sample variances  Determine the value of T from the formula Finding the value of the test statistic Two steps needed: Small Sample Tests of Means            n 1 n s 1 X T p 2 X 2 )1()1(   nn snsn 2  s p

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 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?

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5 H 0 : µ d = µ 1 H 1 : µ d < µ 1  = 0.05 Because both samples are less than 30, the test statistic is t Reject H 0 if t < There are 25 d.f. Pooled Variances 2 )1()1(   nn snsn 2  s p Solution continues…

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Test Compute the test statistic and make a decision Step 5 Pooled Variances 2 )1()1(   nn snsn 2  s p  )9.3)(112( 2  )4.2)(115 2  ( 2  p s  Compute T  =  T   n 1 1 n s 1 X p 2 X  H 0 cannot be rejected… insufficient evidence.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. If you wished to measure the effectiveness of a new diet, you would weigh at the start and at the finish of the programme! If you wished to buy a car, you would look at the same car at two (or more) different dealerships and compare the prices. Hypothesis Testing Involving Paired Observations Examples Independent samples are samples that are not related in any way Dependent samples are samples that are paired or related in same fashion

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. … is the mean of the differences Use the following test when the samples are dependent: Hypothesis Testing Involving Paired Observations D … is the number of pairs (differences) sDsD … is the standard deviation of the differences n T D n  / s D

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 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? CityHertz ($)Avis ($) Halifax$42$40 Quebec City$56$52 Kingston4543 Toronto48 Winnipeg3732 Calgary4548 Saskatoon4139 Vancouver4650

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5 H 0 : µ D = 0  = 0.05 Because both samples are less than 30, the test distribution is t Reject H 0 if T There are 7 d.f. Solution continues… H 1 : µ D    0 1 T D n s D  /

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. DD2D2 Hypothesis Test Compute the test statistic and make a decision Step 5 CityHertz ($)Avis ($) Halifax$42$40 Quebec City$56$52 Kingston4543 Toronto48 Winnipeg3732 Calgary4548 Saskatoon4139 Vancouver Solution continues… T D n s D  /

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Test DD2D n D D    00.1   1  n n  2 D  s D  2 D =     T  Compute the test statistic and make a decision Step 5 Because < 2.365, we can not reject H 0. There is no significant difference in the prices T D n s D  /

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Two Sample Tests of Proportions We investigate whether two samples came from populations with an equal proportion of success. The two samples are pooled using the following formula: X 1 and X 2 refer to the number of successes in the respective samples n 1 and n 2 Test Statistic nn XX ˆ p             21 1 ) ˆ 1( ˆ nn pp  21 ˆˆ pp z

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 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.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5  = 0.05 Because both samples are large the z distribution is used. Reject H 0 if z > Solution continues… H 1 : P u    P m ˆ nn XX p    H 0 : p u p m =

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Test Compute the test statistic and make a decision Step 5        nn  ) ˆ 1( ˆ pp  2 1 ˆˆ p p  z ˆ n n XX p   .1036   250 )1036.1(. 300 ) (.1036      = 1.10

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. = =.1457 The p-value is:P(z > 1.10) Hypothesis Test 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.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Test your learning … Click on… Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s E-Stat data …and much more!

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. This completes Chapter 11