Economics 173 Business Statistics Lecture 9 Fall, 2001 Professor J. Petry

2 Miscellaneous Schedule: today we finish Ch. 12. –Post lecture homework will be assigned this afternoon. Chapter 13 is a review of chapters You should include in your review. Thursday we introduce simple linear regression –Post lecture homework will be assigned after class. Friday you do lab on simple linear regression Next Tuesday, we finish off first part of simple linear regression (approximately through 17.4) and review a few key points in preparation for Thursday’s exam. Mid-term is on Thursday, October 4 from 7-9pm in Lincoln Theatre. There will be no class on that day.

3 In this section we discuss how to compare the variability of two populations. In particular, we draw inference about the ratio of two population variances. This question is interesting because: –Variances can be used to evaluate the consistency of processes. –The relationships between variances determine the technique used to test relationships between mean values 12.5 Inferences about the ratio of two variances

4 Point estimator of  1 2 /  2 2 –Recall that S 2 is an unbiased estimator of  2. –Therefore, it is not surprising that we estimate  1 2 /  2 2 by S 1 2 /S 2 2. Sampling distribution for  1 2 /  2 2 –The statistic [ S 1 2 /  1 2 ] / [ S 2 2 /  2 2 ] follows the F distribution. –The test statistic for  1 2 /  2 2 is derived from this statistic.

5 –Our null hypothesis is always H 0 :  1 2 /  2 2 = 1 –Under this null hypothesis the F statistic becomes F = S12/12S12/12 S22/22S22/22 S12S12 S22S22 Testing  1 2 /  2 2

6 (Example 12.1 revisited) In order to perform a test regarding average consumption of calories at people’s lunch in relation to the inclusion of high-fiber cereal in their breakfast, the variance ratio of two samples has to be tested first. Example 12.5 The hypotheses are: H 0 : H 1 :

7 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?

8 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. But which difference between means test to use? equal or unequal variances version???

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

11 Example The President of Tastee Inc., a baby-food producer, claims that his company’s product is superior to that of his leading competitor, because babies gain weight faster with his product. To test this claim, a survey was undertaken. Mothers of newborn babies were asked which baby food they intended to feed their babies. Those who responded Tastee or the leading competitor were asked to keep track of their babies’ weight gains over the next two months. There were 15 mothers who indicated that they would feed their babies Tasteee and 25 who responded that they would feed their babies the product of the leading competitor. Each baby’s weight gain in ounces is recorded in XR Can we conclude that, using weight gain as our criterion, Tastee baby food is indeed superior? 2.Estimate with 95% confidence the difference between the mean weight of the two products. 3.Check to ensure the required conditions are satisfied.

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13 Example (cont’d)

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Inference about the difference between two population proportions In this section we deal with two populations whose data are qualitative. When data are qualitative we can (only) ask questions regarding the proportions of occurrence of certain outcomes. Thus, we hypothesize on the difference p 1 -p 2, and draw an inference from the hypothesis test.

16 Sample 1 Sample size n 1 Number of successes x 1 Sample proportion Sample 1 Sample size n 1 Number of successes x 1 Sample proportion Sampling Distribution of the Difference Between Two sample proportions –Two random samples are drawn from two populations. –The number of successes in each sample is recorded. –The sample proportions are computed. Sample 2 Sample size n 2 Number of successes x 2 Sample proportion Sample 2 Sample size n 2 Number of successes x 2 Sample proportion x n 1 1 ˆ  p 1

17 –The statistic is approximately normally distributed if n 1 p 1, n 1 (1 - p 1 ), n 2 p 2, n 2 (1 - p 2 ) are all equal to or greater than 5. –The mean of is p 1 - p 2. –The variance of is p 1 (1-p 1 ) /n 1 )+ (p 2 (1-p 2 )/n 2 ) Because p 1, p 2, are unknown, we use their estimates instead. Thus, are all equal to or greater than 5.

18 Testing the Difference between Two Population Proportions –We hypothesize on the difference between the two proportions, p 1 - p 2. –There are two cases to consider: Case 1: H 0 : p 1 -p 2 =0 Calculate the pooled proportion Then Case 2: H 0 : p 1 -p 2 =D (D is not equal to 0) Do not pool the data

19 Example 12.7 –A research project employing 22,000 American physicians was conduct to discover whether aspirin can prevent heart attacks. –Half of the participants in the research took aspirin, and half took placebo. –In a three years period,104 of those who took aspirin and 189 of those who took the placebo had had heart attacks. –Is aspirin effective in preventing heart attacks?

20 Solution –Identifying the technique The problem objective is to compare the population of those who take aspirin with those who do not. The data is qualitative (Take/do not take aspirin) The hypotheses test are H 0 : p 1 - p 2 = 0 H 1 : p 1 - p 2 < 0 We identify here case 1 so Population 1 - aspirin takers Population 2 - placebo takers

21 –Solving by hand For a 5% significance level the rejection region is z < -z  = -z.05 = < , so reject the null hypothesis.

22 Example –Random samples from two binomial populations yielded the following statistics =.45 n 1 =100=.39n 2 =100 –Test with alpha = 0.10 to determine whether we can infer that the population proportions differ.

23 Example –The following statistics were calculated =.12 n 1 =400=.16n 2 =400 –Test with alpha = 0.10 to determine whether p 1 is less than p 2.

24 Example –After sampling two binomial populations we found the following. =.368 n 1 =100=.275n 2 =1000 –Can we infer at the 5% significance level that p 1 is greater than p 2 by more than 5%?