Economics 173 Business Statistics Lecture 8 Fall, 2001 Professor J. Petry

2 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.

4 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

5 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

6 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

7 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

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

9 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

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

11 Case II: The two variances are unequal

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

13 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.

14 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 ).

15 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 )

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

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

18 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?

19 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.

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

21 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

22 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

23 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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25 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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