Economics 173 Business Statistics Lecture 7 Fall, 2001 Professor J. Petry

2 Organization of Techniques Keeping track of the different tests we are conducting is best done with the “Decision Tree” and “Summary” provided in Chapter 22 of your book. As we go through the chapters you should be utilizing the decision tree and Summary to do your problems. –You will be given copies of both for the exams. –We will use the version at the end of the book (chapter 22) so you have the same one to use during the mid-term and the final. –The versions we are handing out today, include statistical tables which, as we announced last class will no longer be used in this course. Develop a process to work each problem. My process is... –Read the question at least twice –Ask myself what type of question does this feel like? Parameter? H 1 ? –Go down the decision tree formally

3 Organization of Techniques Example 1: In a recent municipal election the high cost of housing became an important issue. A candidate seeking to unseat an incumbent claimed that the average family spends more than 30% of its annual income on housing. A housing expert was asked to investigate the claim. A random sample of 125 households was asked to report the percentage of household income spent on housing costs. Assuming you were given the data, what technique would you use to determine if the candidate was correct at the 5% significance level? Example 2: The number of internet users is rapidly increasing. A recent survey reveals that there are about 30 million Internet users in North America. Suppose a survey of 200 of these people were asked to report how many hours they spent on the Internet last week. Assuming you were given the data, what technique would you use to estimate with 95% confidence the average amount of time spent by all North Americans on the Internet?

4 Organization of Techniques Example 3: A rock promoter is in the process of deciding whether to book a new band for a rock concert. He knows that this band appeals almost exclusively to teenagers. According to the latest census, there are 400,000 teenagers in the area. Assuming you were provided the data, what technique would you use to estimate the proportion of teenagers who will attend the concert? Example 4: Some traffic experts believe that the major cause of highway collisions is the differing speeds of cars. That is, when some cars are driven slowly while others are driven fast, cars tend to congregate in bunches increasing the probability of accidents. Thus the greater the variation in speeds, the greater the number of collisions that occur. Suppose that one expert believes that when the variance exceeds 18 (mph), the number of accidents will be unacceptably high. Assuming you are provided the data, what technique would you use to test whether the variance in speeds exceeds 18 (mph)?

5 Inference about the Comparison of Two Populations Chapter 12

Introduction Variety of techniques are presented whose objective is to compare two populations. We are interested in: –The difference between two means. –The ratio of two variances. –The difference between two proportions.

7 Two random samples are drawn from the two populations of interest. Because we are interested in the difference between the two means, we build the statistic for each sample. 12.2Inference about the Difference b/n Two Means: Independent Samples

8 î is normally distributed if the (original) population distributions are normal. î is approximately normally distributed if the (original) population is not normal, but the sample size is large.  Expected value of is  1 -  2  The variance of is  1 2 / n 1 +  2 2 / n 2 The Sampling Distribution of

9 If the sampling distribution of is normal or approximately normal we can write: Z can be used to build a test statistic or a confidence interval for  1 -  2

10 Practically, the “Z” statistic is hardly used, because the population variances are not known. ?? Instead, we construct a “t” statistic using the sample “variances” (S 1 2 and S 2 2 ). S22S22 S12S12 t

11 Two cases are considered when producing the t-statistic. –The two unknown population variances are equal. –The two unknown population variances are not equal.

12 Case I: The two variances are equal Example: S 1 2 = 25; S 2 2 = 30; n 1 = 10; n 2 = 15. Then, Calculate the pooled variance estimate by: n 2 = 15 n 1 = 10

13 Construct the t-statistic as follows: Perform a hypothesis test H 0 :     = 0 H 1 :     > 0; or < 0;or 0 Build an interval estimate

14 Case II: The two variances are unequal

15 Run a hypothesis test as needed, or, build an interval estimate

16 Example 12.1 –Do people who eat high-fiber cereal for breakfast consume, on average, fewer calories for lunch than people who do not eat high-fiber cereal for breakfast? –A sample of 150 people was randomly drawn. Each person was identified as a consumer or a non-consumer of high-fiber cereal. –For each person the number of calories consumed at lunch was recorded.

17 Calories consumed at lunch Solution: The data are quantitative. The parameter to be tested is the difference between two means. The claim to be tested is that mean caloric intake of consumers (  1 ) is less than that of non-consumers (  2 ).

18 Identifying the technique –The hypotheses are: H 0 : (  1 -  2 ) = 0 H 1 : (  1 -  2 ) < 0 – To check the relationships between the variances, we use a computer output to find the samples’ standard deviations. We have S 1 = 64.05, and S 2 = It appears that the variances are unequal. – We run the t - test for unequal variances.  1 <  2 )

19 Calories consumed at lunch At 5% significance level there is sufficient evidence to reject the null hypothesis.

20 Solving by hand –The interval estimator for the difference between two means is

21 Example 12.2 –Do job design (referring to worker movements) affect worker’s productivity? –Two job designs are being considered for the production of a new computer desk. –Two samples are randomly and independently selected A sample of 25 workers assembled a desk using design A. A sample of 25 workers assembled the desk using design B. The assembly times were recorded –Do the assembly times of the two designs differs?

22 Assembly times in Minutes Solution The data are quantitative. The parameter of interest is the difference between two population means. The claim to be tested is whether a difference between the two designs exists.

23 The Excel printout P-value of the one tail test P-value of the two tail test Degrees of freedom t - statistic

24 A 95% confidence interval for  1 -  2 is calculated as follows: Thus, at 95% confidence level <  1 -  2 < Notice: “Zero” is included in the interval

25 Checking the required Conditions for the equal variances case (example 12.2) The distributions are not bell shaped, but they seem to be approximately normal. Since the technique is robust, we can be confident about the results. Design A Design B

26 Example from book Random samples were drawn from each of two populations. The data are stored in columns 1 and 2, respectively, in file XR Is there sufficient evidence at the 5% significance level to infer that the mean of population 1 is greater than the mean of population 2?

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