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Confidence Intervals for the Difference between Two Population Means µ 1 - µ 2 : Independent Samples

Confidence Intervals for the Difference between Two Population Means µ 1 - µ 2 : Independent Samples. Section 6.2. 6.2 Confidence Intervals for the Difference between Two Population Means µ 1 - µ 2 : Independent Samples. Two random samples are drawn from the two populations of interest.

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Confidence Intervals for the Difference between Two Population Means µ 1 - µ 2 : Independent Samples

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  1. Confidence Intervals for the Difference between Two Population Means µ1 - µ2: Independent Samples Section 6.2

  2. 6.2 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 .

  3. 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

  4. Estimate using Sampling distribution model for ? Shape? df Sometimes used (not always very good) estimate of the degrees of freedom is min(n1 − 1, n2 − 1). m1-m2

  5. Confidence Interval for m1– m2

  6. Confidence Interval for m1– m2

  7. Example: “Cameron Crazies”. Confidence interval for m1– m2 • Do the “Cameron Crazies” at Duke home games help the Blue Devils play better defense? • Below are the points allowed by Duke (men) at home and on the road for the conference games from a recent season.

  8. Example: “Cameron Crazies”. Confidence interval for m1– m2 Calculate a 95% CI for 1 - 2where 1 = mean points per game allowed by Duke at home. 2= mean points per game allowed by Duke on road • n1 = 8, n2 = 8; s12= (21.8)2= 475.36; s22 = (8.9)2 = 79.41

  9. Example: “Cameron Crazies”. Confidence interval for m1– m2 • To use the t-table let’s use df = 9; t9* = 2.2622 • The confidence interval estimator for the difference between two means is …

  10. Interpretation • The 95% CI for 1 - 2 is (-19.22, 18.46). • Since the interval contains 0, there appears to be no significant difference between 1 = mean points per game allowed by Duke at home. 2= mean points per game allowed by Duke on road • The Cameron Crazies appear to have no affect on the ABILITY of the Duke men to play defense. How can this be?

  11. Example: confidence interval for m1– m2 • Example (p. 6) • 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.

  12. Example: confidence interval for m1– m2 Solution: (all data on p. 6) • 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). • n1 = 43, n2 = 107; s12=4,103; s22=10,670

  13. Example: confidence interval for m1– m2 • Let’s use df = 120; t120* = 1.9799 • The confidence interval estimator for the difference between two means using the formula on p. 4 is

  14. Interpretation • The 95% CI is (-56.87, -1.55). • 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 1.55 and 56.87 more calories for lunch.

  15. Example: (cont.) confidence interval for 1 – 2using min(n1 –1, n2 -1) to approximate the df • Let’s use df = min(43-1, 107-1) = min(42, 106) = 42; • t42* = 2.0181 • The confidence interval estimator for the difference between two means using the formula on p. 4 is

  16. Beware!! Common Mistake !!! A common mistake is to calculate a one-sample confidence interval for m1, a one-sample confidence interval for m2, and to then conclude that m1and m2 are equal if the confidence intervals overlap. This is WRONG because the variability in the sampling distribution for from two independent samples is more complex and must take into account variability coming from both samples. Hence the more complex formula for the standard error.

  17. INCORRECT Two single-sample 95% confidence intervals: The confidence interval for the male mean and the confidence interval for the female mean overlap, suggesting no significant difference between the true mean for males and the true mean for females. Male interval: (18.68, 20.12) Female interval: (16.94, 18.86) 0 1.5 .313 2.69

  18. Reason for Contradictory Result

  19. 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?

  20. 95% confidence interval for (µ1 − µ2), with df = min(30-1, 30-1) = 29 t* = 2.0452: • 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 is between 19.33 and 6.07 milliliters LESS in children of smoking parents.

  21. 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?

  22. 95% confidence interval for (µ1 − µ2), with df = min(99-1, 888-1) = 98 t* = 1.9845: The “Bambino”,left-handed Babe Ruth, baseball’s all-time best player. • 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 3.26 and 13.54 years LESS than the mean life expectancy for right-handers.

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