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Confidence Intervals for µ 1 - µ 2 and p 1 - p 2

This article discusses confidence intervals for the difference between two population means and proportions. Examples are provided to illustrate the calculation and interpretation of confidence intervals.

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Confidence Intervals for µ 1 - µ 2 and p 1 - p 2

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  1. Confidence Intervals for µ1- µ2 and p1 - p2

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

  3. Confidence Intervals for the Difference between Two Population Means µ1 - µ2: Independent Samples • Two random samples are drawn from the two populations of interest. • Because we compare two population means, we use the statistic .

  4. Population 1Population 2 Parameters: µ1 and 12Parameters: µ2 and 22 (values are unknown) (values are unknown) Sample size: n1Sample size: n2 Statistics: x1 and s12Statistics: x2 and s22 Estimate µ1 µ2 with x1 x2

  5. Confidence Interval for m1– m2 Note: when the values of 12 and 22 are unknown, the sample variances s12 and s22 computed from the data can be used.

  6. Example: confidence interval for m1– m2 • 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.

  7. Example: confidence interval for m1– m2 • Solution: • The parameter to be tested is • the difference between two means. • The claim to be tested is: • The mean caloric intake of consumers (m1) • is less than that of non-consumers (m2). • Use s12 = 4,103 for 12 and s22 = 10,670 • for 22

  8. Example: confidence interval for m1– m2 • The confidence interval estimator for the difference between two means is

  9. 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 m1– m2 • Since the interval is entirely negative (that is, does not contain 0), there is evidence from the data that µ1 is less than µ2. We estimate that non-consumers of high-fiber breakfast consume on average between 1.83 and 56.59 more calories for lunch.

  10. 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?

  11. 95% confidence interval for (µ1 − µ2): • 1 = mean FVC of children with a smoking parent; • 2 = mean FVC of children without a smoking parent We are 95% confident that lung capacity in children of smoking parents is between 19.05 and 6.35 milliliters LESS than in children without a smoking parent.

  12. 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. 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.

  13. 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.

  14. 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. left-handed presidents star left-handed quarterback Steve Young 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?

  15. 95% confidence interval for (µ1 − µ2): The “Bambino”,left-handed hitter Babe Ruth, baseball’s all-time best hitter • 1 = mean life expectancy of left-handers; • 2 = mean life expectancy of right-handers We are 95% confident that the mean life expectancy for left-handers is between 3.32 and 13.48 years LESS than the mean life expectancy for right-handers.

  16. Example: confidence interval for m1– m2 • 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.

  17. Example: confidence interval for m1– m2 • 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?

  18. Example: confidence interval for m1– m2 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 s12 = .848 for 12 and s22 = 1.303 • for 22

  19. Example: confidence interval for m1– m2 A 95% confidence interval for m1 - m2 is calculated as follows: We are 95% confident that the interval (-0.3029 , 0.8469) contains the true but unknown m1 - m2 Notice: “Zero” is included in the confidence interval

  20. Confidence Intervals for the difference p1 – p2 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

  21. 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 p1 – p2. • Statistic • An estimator of p1 – p2 is (the difference between the sample proportions).

  22. Point Estimator: x ˆ = p 1 1 n 1 • Two random samples are drawn from two populations. • The number of successes in each sample is recorded. • The sample proportions are computed. Sample 1 Sample size n1 Number of successes x1 Sample proportion Sample 2 Sample size n2 Number of successes x2 Sample proportion

  23. Confidence Interval for p1 p2

  24. Example: confidence interval for p1 – p2 • 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.

  25. Example: confidence interval for p1 – p2(cont.) • 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 p1 – p2. • p1 = death rate with Streptokinase • p2 = death rate with t-PA

  26. Example: confidence interval for p1 – p2(cont.) • 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.

  27. Example: confidence interval for p1 – p2(cont.) • Compute: Manually • Sample proportions: • The 95% confidence interval estimate is

  28. Example: confidence interval for p1 – p2(cont.) • Interpretation • The interval (.0051, .0149) for p1 – p2 does not contain 0; it is entirely positive, which indicates that p1, the death rate for streptokinase, is greater than p2, 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.

  29. Example: 95% confidence interval for p1 – p2 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 30th birthday or after. The parameter to be estimated is p1 – p2. p1 = cancer rate when age at 1st birth >30 p2 = cancer rate when age at 1st birth <=30 We estimate that the cancer rate when age at first birth > 30 is between .05 and .082 higher than when age <= 30.

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