1 Selected Sections of Chapters 22 and 24 Confidence Intervals for p 1 - p 2 and µ 1 - µ 2

2 Inference about Two Populations We are interested in: –Confidence intervals for the difference between two proportions. –Confidence intervals for the difference between two means.

3 Confidence Intervals for the difference p 1 – p 2 between two population proportions In this section we deal with two populations whose data are qualitative. For qualitative data we compare the population proportions of the occurrence of a certain event. Examples –Comparing the effectiveness of new drug versus older one –Comparing market share before and after advertising campaign –Comparing defective rates between two machines

4 Parameter and Statistic Parameter –When the data are qualitative, we can only count the occurrences of a certain event in the two populations, and calculate proportions. – The parameter we want to estimate is p 1 – p 2. Statistic –An estimator of p 1 – p 2 is (the difference between the sample proportions).

5 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 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 Point Estimator:

6 Confidence Interval for p 1  p 2

7 Example: confidence interval for p 1 – p 2 Estimating the cost of life saved –Two drugs are used to treat heart attack victims: Streptokinase (available since 1959, costs $460) t-PA (genetically engineered, costs $2900). –The maker of t-PA claims that its drug outperforms Streptokinase. –An experiment was conducted in 15 countries. 20,500 patients were given t-PA 20,500 patients were given Streptokinase The number of deaths by heart attacks was recorded.

8 Experiment results –A total of 1497 patients treated with Streptokinase died. –A total of 1292 patients treated with t-PA died. Estimate the difference in the death rates when using Streptokinase and when using t-PA. Example: confidence interval for p 1 – p 2 (cont.)

9 Solution –The problem objective: Compare the outcomes of two treatments. –The data are qualitative (a patient lived or died) –The parameter to be estimated is p 1 – p 2. p 1 = death rate with Streptokinase p 2 = death rate with t-PA Example: confidence interval for p 1 – p 2 (cont.)

10 Compute: Manually –Sample proportions: –The 95% confidence interval estimate is Example: confidence interval for p 1 – p 2 (cont.)

11 Interpretation –The interval (.0051,.0149) for p 1 – p 2 does not contain 0; it is entirely positive, which indicates that p 1, the death rate for streptokinase, is greater than p 2, the death rate for t-PA. –We estimate that the death rate for streptokinase is between.51% and 1.49% higher than the death rate for t-PA. Example: confidence interval for p 1 – p 2 (cont.)

12 Example: 95% confidence interval for p 1 – p 2 The age at which a woman gives birth to her first child may be an important factor in the risk of later developing breast cancer. An international study conducted by WHO selected women with at least one birth and recorded if they had breast cancer or not and whether they had their first child before their 30 th birthday or after. CancerSample Size Age at First Birth > % Age at First Birth <= , % The parameter to be estimated is p1 – p2. p1 = cancer rate when age at 1 st birth >30 p2 = cancer rate when age at 1 st birth <=30 We estimate that the cancer rate when age at first birth > 30 is between.05 and.082 higher than when age <= 30.

13 Two random samples are drawn from the two populations of interest. Because we compare two population means, we use the statistic. Confidence Intervals for the Difference between Two Population Means µ 1 - µ 2 : Independent Samples

14 Population 1Population 2 Parameters: µ 1 and  1 2 Parameters: µ 2 and  2 2 (values are unknown) (values are unknown) Sample size: n 1 Sample size: n 2 Statistics: x 1 and s 1 2 Statistics: x 2 and s 2 2 Estimate µ 1  µ 2 with x 1  x 2

15 Confidence Interval for    –   Note: when the values of  1 2 and  2 2 are unknown, the sample variances s 1 2 and s 2 2 computed from the data can be used.

16 –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. Example: confidence interval for    –  

17 Solution: The parameter to be tested is the difference between two means. The claim to be tested is: The mean caloric intake of consumers (  1 ) is less than that of non-consumers (  2 ). Use s 1 2 = 4,103 for  1 2 and s 2 2 = 10,670 for  2 2 Example: confidence interval for    –  

18 The confidence interval estimator for the difference between two means is Example: confidence interval for    –  

19 Interpretation The 95% CI is (-56.59, -1.83). We are 95% confident that the interval (-56.59, -1.83) contains the true but unknown difference    –   Since the interval is entirely negative (that is, does not contain 0), there is evidence from the data that µ 1 is less than µ 2. We estimate that non-consumers of high-fiber breakfast consume on average between 1.83 and more calories for lunch.

Does smoking damage the lungs of children exposed to parental smoking? Forced vital capacity (FVC) is the volume (in milliliters) of air that an individual can exhale in 6 seconds. FVC was obtained for a sample of children not exposed to parental smoking and a group of children exposed to parental smoking. We want to know whether parental smoking decreases children’s lung capacity as measured by the FVC test. Is the mean FVC lower in the population of children exposed to parental smoking? Parental smokingFVCsn Yes No

Parental smokingFVCsn Yes No We are 95% confident that lung capacity in children of smoking parents is between and 6.35 milliliters LESS than in children without a smoking parent. 95% confidence interval for (µ 1 − µ 2 ):  1 = mean FVC of children with a smoking parent;  2 = mean FVC of children without a smoking parent

Bunny Rabbits and Pirates on the Box The data below show the sugar content (as a percentage of weight) of 10 brands of cereal randomly selected from a supermarket shelf that is at a child’s eye level and 8 brands selected from the top shelf. 22 Eye level Top Create and interpret a 95% confidence interval for the difference  1 –  2 in mean sugar content, where  1 is the mean sugar content of cereal at a child’s eye level and  2 is the mean sugar content of cereal on the top shelf.

Eye level Top

Interpretation We are 95% confident that the interval (28.46, 40.22) contains the true but unknown value of  1 –  2. Note that the interval is entirely positive (does not contain 0); therefore, it appears that the mean amount of sugar  1 in cereal on the shelf at a child’s eye level is larger than the mean amount  2 on the top shelf. 24

Do left-handed people have a shorter life-expectancy than right-handed people?  Some psychologists believe that the stress of being left- handed in a right-handed world leads to earlier deaths among left-handers.  Several studies have compared the life expectancies of left- handers and right-handers.  One such study resulted in the data shown in the table. We will use the data to construct a confidence interval for the difference in mean life expectancies for left- handers and right-handers. Is the mean life expectancy of left-handers less than the mean life expectancy of right-handers? HandednessMean age at deathsn Left Right left-handed presidents star left-handed quarterback Steve Young

We are 95% confident that the mean life expectancy for left- handers is between 3.32 and years LESS than the mean life expectancy for right-handers. 95% confidence interval for (µ 1 − µ 2 ):  1 = mean life expectancy of left-handers;  2 = mean life expectancy of right-handers HandednessMean age at deathsn Left Right The “Bambino”,left-handed hitter Babe Ruth, baseball’s all-time best hitter

27 Example –An ergonomic chair can be assembled using two different sets of operations (Method A and Method B) –The operations manager would like to know whether the assembly time under the two methods differ. Example: confidence interval for    –  

28 Example –Two samples are randomly and independently selected A sample of 25 workers assembled the chair using method A. A sample of 25 workers assembled the chair using method B. The assembly times were recorded –Do the assembly times of the two methods differs? Example: confidence interval for    –  

29 Example: confidence interval for    –   Assembly times in Minutes Solution The parameter of interest is the difference between two population means. The claim to be tested is whether a difference between the two methods exists. Use s 1 2 =.848 for  1 2 and s 2 2 = for  2 2

30 Example: confidence interval for    –   A 95% confidence interval for  1 -  2 is calculated as follows: We are 95% confident that the interval ( , ) contains the true but unknown  1 -  2 Notice: “Zero” is included in the confidence interval