© 2012 McGraw-Hill Ryerson Limited1 © 2009 McGraw-Hill Ryerson Limited

© 2012 McGraw-Hill Ryerson Limited2 Lind Marchal Wathen Waite

© 2012 McGraw-Hill Ryerson Limited3 Conduct a test of a hypothesis about the difference between two independent population means. Conduct a test of a hypothesis about the difference between two population proportions. Conduct a test of a hypothesis about the mean difference between paired or dependent observations. Explain the difference between dependent and independent samples. Learning Objectives LO

© 2012 McGraw-Hill Ryerson Limited4 Introduction

© 2012 McGraw-Hill Ryerson Limited5 Is there a difference in the mean value of residential real estate sold by male agents and female agents in Halifax? Is there a difference in the mean number of defects produced on the day and the afternoon shifts at Kimble Products? Is there a difference in the mean number of days absent between young workers (under 21 years of age) and older workers (more than 60 years of age) in the fast food industry? Is there an increase in the production rate if music is piped into the production area? Comparing Two Populations Examples

© 2012 McGraw-Hill Ryerson Limited6 Two-Sample Tests of Hypothesis: Independent Samples LO 1

© 2012 McGraw-Hill Ryerson Limited7 Assume that a distribution of sample means will follow the normal distribution. It can be shown mathematically that the distribution of the differences between sample means for two normal distributions is also normal. If we find the mean of this distribution is 0, that implies that there is no difference in the two populations. If the mean of the distribution of differences is equal to some value other than 0, either positive or negative, then we conclude that the two populations do not have the same mean. Two-Sample Tests of Hypothesis: Independent Samples LO 1

© 2012 McGraw-Hill Ryerson Limited8 The distribution of the differences has a variance (standard deviation squared) equal to the sum of the two individual variances. This means that we can add the variances of the two sampling distributions. Variability of the Distribution of Differences LO 1

© 2012 McGraw-Hill Ryerson Limited9 1.The two samples must be unrelated, that is, independent. 2.The standard deviations of both populations must be known. Two-Sample Tests of Means – Standard Deviation Known LO 1

© 2012 McGraw-Hill Ryerson Limited10 An Automatic Payment Machine (APM) has been installed in Link Road Super market. Customers can make payment by cash as well as by debit or credit card with the help of Automatic Payment Machine (APM). The store manager would like to know if the mean time using the cash payment method is longer than using the automatic payment machine. The time is measured from when the customer enters the line until his or her payments are made. Example – Two-Sample Tests of Means – Standard Deviation Known LO 1

© 2012 McGraw-Hill Ryerson Limited11 Hence, the time includes both waiting in line and checking out. What is the p-value? Example – Two-Sample Tests of Means – Standard Deviation Known Method used for Payment Sample Mean Population Standard Deviation Sample Size Cash5.50 minutes 0.40 minutes 50 APM5.30 minutes0.30 minutes100 LO 1 Continued

© 2012 McGraw-Hill Ryerson Limited12 Step 1: State the null hypothesis and the alternate hypothesis. H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Step 2: Select the level of significance. α = 0.01, as selected by researcher Step 3: Select the test statistic. Because both population standard deviations are given, we use z distribution as the test statistic. Solution – Two-Sample Tests of Means – Standard Deviation Known LO 1

© 2012 McGraw-Hill Ryerson Limited13 Step 4: Formulate the decision rule. Reject H 0 if z exceeds Decision Rule for One-Tailed Test at.01 Significance Level Solution – Two-Sample Tests of Means – Standard Deviation Known LO 1 Continued

© 2012 McGraw-Hill Ryerson Limited14 Step 5: Make a decision and interpret the result. The computed value of 3.13 is larger than the critical value of Our decision is to reject the null hypothesis and accept the alternative hypothesis. The difference of 0.20 minutes between the mean time using the cash payment method is too large to have occurred by chance. To put it another way, we conclude that the payment made using the (APM) is faster. Solution – Two-Sample Tests of Means – Standard Deviation Known LO 1 Continued

© 2012 McGraw-Hill Ryerson Limited15 Two-Sample Tests of Means – Standard Deviation Known In Excel LO 1

© 2012 McGraw-Hill Ryerson Limited16 You Try It Out! The mean time to solve 10 problems by method A is 300 seconds. The mean time to solve 12 similar problems by method B is 280 seconds. The population standard deviation is 35 seconds for method A and 29 seconds for method B. At the.05 significance level, can we conclude that the mean time required to solve the problems by method A is greater than method B? LO 1

© 2012 McGraw-Hill Ryerson Limited17 You Try It Out! (a)State the null hypothesis and the alternative hypothesis. (b)What is the decision rule? (c)What is the value of the test statistic? (d)What is your decision regarding the null hypothesis? (e)What is the p-value? (f )Interpret the result. LO 1

© 2012 McGraw-Hill Ryerson Limited18 Two-Sample Tests About Proportions LO 2

© 2012 McGraw-Hill Ryerson Limited19 1)Many Canadians wish to know whether there is a difference in the proportion of Conservatives and New Democrats who favour creating public service jobs in Canada. 2)A stylist at The Hair Care Palace has invented a new hair dye colour. The new colour is shown to a group of customers under 30 years of age and to another group over 50 years of age. The stylist wants to know if there is a difference in the proportion of the two groups who like the new hair colour. Two-Sample Tests About Proportions–Examples LO 2

© 2012 McGraw-Hill Ryerson Limited20 3)A consultant in a financial firm is investigating the income rate. Specifically, she wishes to know whether there is a difference in the proportion of men versus women with respect to income. Two-Sample Tests About Proportions–Examples LO 2

© 2012 McGraw-Hill Ryerson Limited21 We assume the nominal scale of measurement. To conduct the test, we assume each sample is large enough that the normal distribution will serve as a good approximation of the binomial distribution. The test statistic follows the standard normal distribution. Two-Sample Tests About Proportions LO 2

© 2012 McGraw-Hill Ryerson Limited22 Formula (10–3) is formula (10–2) with the respective sample proportions replacing the sample means and replacing the two variances. In addition: n 1 is the number of observations in the first sample. n 2 is the number of observations in the second sample. is the proportion in the first sample possessing the trait. is the proportion in the second sample possessing the trait. Two-Sample Tests About Proportions LO 2

© 2012 McGraw-Hill Ryerson Limited23 is the pooled proportion possessing the trait in the combined samples. It is called the pooled estimate of the population proportion and is computed from the following formula. where: X 1 is the number possessing the trait in the first sample. X 2 is the number possessing the trait in the second sample. Two-Sample Tests About Proportions LO 2

© 2012 McGraw-Hill Ryerson Limited24 A student government representative at a local university wanted to study whether there is a difference in the proportion of male students and female students who choose engineering branch for further studies. A random sample of 100 male students revealed 19 will choose engineering branch for further studies. Similarly, a sample of 200 female students revealed 62 will choose engineering branch for further studies. Example – Two-Sample Tests About Proportions LO 2

© 2012 McGraw-Hill Ryerson Limited25 Step 1: State the null hypothesis and the alternate hypothesis. H 0 : p 1 – p 2 = 0 H 1 : p 1 – p 2 ≠ 0 Step 2: Select the level of significance. α = 0.05 Step 3: Select the test statistic. The test statistic follows the standard normal distribution and the test statistic can be computed from formula Solution – Two-Sample Tests About Proportions LO 2

© 2012 McGraw-Hill Ryerson Limited26 Step 4: Formulate the decision rule. Reject H 0 if z is not between –1.96 and Decision Rules for.05 Significance Level Solution – Two-Sample Tests About Proportions LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited27 Step 5: Make a decision and interpret the result. The computed value of –2.21 is in the area of rejection; that is, it is to the left of –1.96. Therefore, the null hypothesis is rejected at the 0.05 significance level. Solution – Two-Sample Tests About Proportions LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited28 Solution – Two-Sample Tests About Proportions LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited29 To find the p-value, we go to Appendix D and look for the likelihood of finding a z-value less than –2.21 or greater than The z value corresponding to 2.21 is , so the likelihood of finding the value of the test statistic to be less than –2.21 or greater than 2.21 is: p-value = 2( – ) = The p-value of is less than the significance level of 0.05, so our decision is to reject the null hypothesis. Again, we conclude that there is a difference in the proportion of male students and female students who choose engineering branch for further studies. Solution – Two-Sample Tests About Proportions LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited30 Two-Sample Tests About Proportions in Excel LO 2

© 2012 McGraw-Hill Ryerson Limited31 You Try It Out! Of 200 girls, 165 wanted to go to camping for the weekend. And of 210 boys, 186 boys wanted to go camping. Using the.10 level of significance, can we conclude that there is a significant difference in the proportion of girls and the proportion of boys who preferred camping? LO 2

© 2012 McGraw-Hill Ryerson Limited32 You Try It Out! (a)State the null hypothesis and the alternative hypothesis. (b)What is the probability of a Type I error? (c)Is this a one-tailed or a two-tailed test? (d)What is the decision rule? (e)What is the value of the test statistic? (f)What is your decision regarding the null hypothesis? (g)What is the p-value? Explain what it means in terms of this problem. LO 2

© 2012 McGraw-Hill Ryerson Limited33 Major differences from previous section: 1.The sampled populations have equal but unknown standard deviations Because of this assumption we combine or “pool” the sample standard deviations. 2.We use the t distribution as the test statistics Requirements include: 3.Each population follows a normal distribution. 4.The sampled populations are independent. 5.The standard deviations of the two populations are equal. Comparing Population Means with Unknown Population Standard Deviations (the Pooled t-test) LO 2

© 2012 McGraw-Hill Ryerson Limited34 where: is the variance (standard deviation squared) of the first sample. is the variance of the second sample. Comparing Population Means with Unknown Population Standard Deviations (the Pooled t-test) LO 2

© 2012 McGraw-Hill Ryerson Limited35 where: is the mean of the first sample. is the mean of the second sample. n 1 is the number of observations in the first sample. n 2 is the number of observations in the second sample. is the pooled estimate of the population variance. Comparing Population Means with Unknown Population Standard Deviations (the Pooled t-test) LO 2

© 2012 McGraw-Hill Ryerson Limited36 A city park has a circular bicycle path 2 km long. Five people rode along the path on Wells bicycles and six people rode it on Atkins bicycles. Is there a difference in the mean riding time with these two different makes of bicycles? To evaluate the two methods, it was decided to conduct a time and motion study. Five people were timed using the Wells bicycle and six using the Atkins bicycle. Example – Comparing Population Means with Unknown Population Standard Deviations LO 2

© 2012 McGraw-Hill Ryerson Limited37 Is there a difference in the mean riding times? Use the 0.10 significance level. Example – Comparing Population Means with Unknown Population Standard Deviations Wells (minutes)Atkins (minutes) LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited38 Step 1: State the null hypothesis and the alternate hypothesis. H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0 Step 2: Select the level of significance. α = 0.10 Step 3: Select the test statistic. Because the population standard deviations are not known but are assumed to be equal, we use the pooled t-test. Solution – Comparing Population Means with Unknown Population Standard Deviations LO 2

© 2012 McGraw-Hill Ryerson Limited39 Step 4: Formulate the decision rule. Reject H 0 if t is not between –1.833 and Regions of Rejection, Two-Tailed Test, df = 9, and.10 Significance Level Solution – Comparing Population Means with Unknown Population Standard Deviations LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited40 Step 5: Make a decision and interpret the result. The decision is not to reject the null hypothesis. We conclude that there is no difference in the mean riding time with these two different makes of bicycles. Solution – Comparing Population Means with Unknown Population Standard Deviations LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited41 We use three steps to compute the value 0 of t. Find the sample standard deviations. Solution – Comparing Population Means with Unknown Population Standard Deviations WellsAtkins X1X1 X12X12 X2X2 X22X LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited42 Pool the sample variances. Determine the value of t. Solution – Comparing Population Means with Unknown Population Standard Deviations LO 2 Continued

© 2012 McGraw-Hill Ryerson Limited43 Comparing Population Means with Unknown Population Standard Deviations in Excel LO 2

© 2012 McGraw-Hill Ryerson Limited44 Comparing Population Means with Unknown Population Standard Deviations in Excel LO 2

© 2012 McGraw-Hill Ryerson Limited45 You Try It Out! The number of defective monitors manufactured in the day shifts and afternoon shifts are to be compared. A sample of the production from six day shifts and eight afternoon shifts showed the following number of defects. Is there a difference in the mean number of defects per shift? Choose an appropriate significance level. LO 2 Day Afternoon

© 2012 McGraw-Hill Ryerson Limited46 You Try It Out! (a)State the null hypothesis and the alternative hypothesis. (b)What is the decision rule? (c)What is the value of the test statistic? (d)What is your decision regarding the null hypothesis? (e)What is the p-value? (f )Interpret the result. (g)What assumptions are necessary for this test? LO 2

© 2012 McGraw-Hill Ryerson Limited47 Two-Sample Tests of Hypothesis: Dependent Samples LO 3

© 2012 McGraw-Hill Ryerson Limited48 Dependent samples are samples that are paired or related in some fashion. For example: 1)If you wished to buy a car you would look at the same car at two (or more) different dealerships and compare the prices. 2)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. Two-Sample Tests of Hypothesis: Dependent Samples LO 3

© 2012 McGraw-Hill Ryerson Limited49 There are n – 1 degrees of freedom and is the mean of the difference between the paired or related observations. S d is the sample standard deviation of the differences between the paired or related observations. nis the number of paired observations. Two-Sample Tests of Hypothesis: Dependent Samples LO 3

© 2012 McGraw-Hill Ryerson Limited50 Schadek and Bowyer are two construction and development firms bought some properties for development. A sample of 10 properties for both firms has been selected. The price of properties, given in thousands of dollars, ($000) are shown in the table. At the.05 significance level, can we conclude there is a difference in the mean property value of both the firms? Example – Two-Sample Tests of Hypothesis: Dependent Samples Home Schadek ($ thousands) Bowyer ($ thousands) 1$235$ LO 3

© 2012 McGraw-Hill Ryerson Limited51 Step 1: State the null hypothesis and the alternate hypothesis. H 0 : μ d = 0 H 1 : μ d ≠ 0 Step 2: Select the level of significance. α = 0.05 Step 3: Select the test statistic. We will use the t-test. Solution – Two-Sample Tests of Hypothesis: Dependent Samples LO 3

© 2012 McGraw-Hill Ryerson Limited52 Step 4: Formulate the decision rule. Reject H 0 if t is not between –2.262 and Solution – Two-Sample Tests of Hypothesis: Dependent Samples df 80%90%95%98%99%99.90% Level of Significance for One-Tailed Test Level of Significance for Two-Tailed Test LO 3 Continued

© 2012 McGraw-Hill Ryerson Limited53 Step 5: Make a decision and interpret the result. The null hypothesis is rejected. The population distribution of differences does not have a mean of 0. We conclude that there is a difference in the mean property value of both the firms. Solution – Two-Sample Tests of Hypothesis: Dependent Samples Home SchadekBowyer Difference, d Difference ($ thousands) Squared, d 2 1$235$ LO 3 Continued

© 2012 McGraw-Hill Ryerson Limited54 Using formula (10-7), the value of the test statistic is 3.305, found by Solution – Two-Sample Tests of Hypothesis: Dependent Samples LO 3 Continued

© 2012 McGraw-Hill Ryerson Limited55 Two-Sample Tests of Hypothesis: Dependent Samples in Excel LO 3

© 2012 McGraw-Hill Ryerson Limited56 Two-Sample Tests of Hypothesis: Dependent Samples in Excel LO 3

© 2012 McGraw-Hill Ryerson Limited57 Comparing Dependent and Independent Samples LO 4

© 2012 McGraw-Hill Ryerson Limited58 Two types of dependent samples: 1.Those characterized by a measurement, an intervention of some type, then another measurement, called a “before” and “after” study. 2.A matching or pairing of the observations. Comparing Dependent and Independent Samples LO 4

© 2012 McGraw-Hill Ryerson Limited59 Preference is for dependent samples: 1.able to reduce the variation in the sampling distribution. 2.its standard error is always smaller. 3.leads to a larger test statistic and a greater chance of rejecting the null hypothesis. 4.bad news is reduced degrees of freedom, but a small price to pay for a better test. Comparing Dependent and Independent Samples LO 4

© 2012 McGraw-Hill Ryerson Limited60 You Try It Out! A fitness centre announces a guaranteed weight-loss program. A random sample of eight recent participants showed the following weights before and after completing the course. At the 0.01 significance level, can we conclude the participants did lose weight? LO 4 NamesBeforeAfter Hunter Cashman Mervine Massa Creola Perterson Redding Poust185175

© 2012 McGraw-Hill Ryerson Limited61 You Try It Out! ( a)State the null hypothesis and the alternative hypothesis. (b)What is the critical value of t? (c)What is the computed value of t? (d)Interpret the result. What is the p-value? (e)What assumption needs to be made about the distribution of the differences? LO 4

© 2012 McGraw-Hill Ryerson Limited62 Chapter Summary I.In comparing two population means we wish to know whether they could be equal. A.We are investigating whether the distribution of the difference between the means could have a mean of 0. B.The test statistic follows the standard normal distribution if the population standard deviations are known. 1.No assumption about the shape of either population is required. 2.The samples are from independent populations.

© 2012 McGraw-Hill Ryerson Limited63 Chapter Summary 3.The formula to compute the value of z is: [10–2] II.We can also test whether two samples came from populations with an equal proportion of successes. A.The two-sample proportions are pooled using the following formula: [10–4]

© 2012 McGraw-Hill Ryerson Limited64 Chapter Summary B.We compute the value of the test statistic from the following formula: [10–3] III.The test statistic to compare two means is the t distribution if the population standard deviations are not known. A.Both populations must follow the normal distribution. B.The populations must have equal standard deviations. C.The samples are independent.

© 2012 McGraw-Hill Ryerson Limited65 Chapter Summary D.Finding the value of t requires two steps. 1.First, pool the sample standard deviations according to the following formula: [10–5] 2.Then compute the value of t from the following formula: [10–6] 3.The degrees of freedom for the test are n 1 + n 2 – 2.

© 2012 McGraw-Hill Ryerson Limited66 Chapter Summary IV.For dependent samples, we assume the distribution of the paired differences between the populations has a mean of 0. A.We first compute the mean and the standard deviation of the sample differences. B.The value of the test statistic is computed from the following formula: [10–7]