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Confidence Interval of a Mean

Confidence Interval of a Mean. FPP 23. Confidence intervals for proportion review. Generic formula for a confidence interval estimate ± multiplier*SE Recall the multiplier depends on the level of confidence For a population proportion we have

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Confidence Interval of a Mean

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  1. Confidence Interval of a Mean FPP 23

  2. Confidence intervals for proportion review • Generic formula for a confidence interval • estimate ± multiplier*SE • Recall the multiplier depends on the level of confidence • For a population proportion we have • The multiplier here is found using the normal distribution

  3. Confidence interval for a mean • Generic formula • estimate ± multiplier*SE • An estimate for a population mean μis the sample mean (typically denoted by ) • SE is given by σ/√n • Multiplier found using the normal distribution • But we don’t know σ. So what do we do? • Use the sample standard deviation • Thus • But since we use s instead of σ we must use a t-distribution with n – 1 degrees of freedom (d.f.) instead of a normal distribution to find the multiplier

  4. t-table

  5. CI of a mean recap • Equation for a confidence interval of a mean • sample mean ± multiplier*SE • The multiplier comes from the t-distribution with n – 1 d.f., s is the sample standard deviation, n is the sample size • All the ideas of confidence intervals for a proportion carry over to means. • Interpretations • The meaning of statistical confidence.

  6. Application of CI’s: Mercury levels in NC rivers • Rivers in North Carolina contain small concentrations of mercury which can accumulate in fish over their lifetimes. Because mercury cannot be excreted from the body it builds up in the tissues. The concentration of mercury in fish tissues can be obtained at considerable expense by catching fish and sending samples to a lab for analysis. Directly measuring the mercury concentration in the water is impossible since it is almost always below detectable limits • A study was recently conducted by researchers at the Nicholas School of the Environment at Duke in the Wacamaw and Lumber Rivers to investigate mercury levels in tissues of large mouth bass. At several stations along each river, a group of fish were caught, weighted and measured. In addition a filet from each fish caught was sent to the lab so that the tissue concentration of mercury (in parts per million) could be determined for each fish. • Mercury in concentrations greater than 1 part per million are considered unsafe for humans to ingest. Are fish in the Lumber and Wacamaw Rivers too contaminated to eat?

  7. EDA for mercury • The distribution of mercury is right-skewed in both rivers. There are a few outliers in Lumber River, but the large sample size should allow us to use the Central Limit Theorem for CI’s. • The sample average mercury level for both rivers is above 1.0 ppm. • 95% CI’s for population average mercury levels in two rivers:

  8. Conclusions based on CI’s • We are 95% confident that the population average mercury level in fish in the Lumber River is between .93 and 1.23 ppm. Since 1.0 ppm is inside the CI, we do not feel confident that the average level is below or above the danger level. More study is needed. • We are 95% confident that the population average mercury level in fish in the Wacamaw River is between 1.11 and 1.44 ppm. It is likely that the average mercury level is beyond 1.0 ppm and therefore unsafe. Don’t eat Wacamaw bass!

  9. Interpretation of CI’s for averages • Wrong: • “95% of all fish in Wacamaw river have mercury levels between 1.11 and 1.44 pm” • Right • “We are 95% confident that the average mercury level of fish in the Wacamaw river is between 1.11 and 1.44ppm”

  10. Special consideration for CI’s of averages • Beware of outliers • Outliers can dramatically inflate estimates of the SE. This could lead to CI’s so wide they aren’t useful. • What to do when you have outliers: • Check for data entry errors • Do analyses with and without outliers. When results differ substantially, report both analyses. Otherwise, report original analyses only.

  11. Example 1 • Suppose Brent Matthews, manager of a Sam’s Club, wants to know how much milk he should stock daily. Brent checked the sales records for random sample of 16 days and found the mean number of gallons sold is 150 gallons per day, the sample standard deviation is 12 gallons. Determine the number of gallons that Brent should stock daily with a 95% confidence interval.

  12. Example 2 • It is important for airlines to follow the published scheduled departure times of flights. Suppose that one airline that recently sampled the records of 246 flights originating in Orlando found that 10 flights were delayed for severe weather, 4 flights were delayed for maintenance concerns, and all the other flights were on time. Determine the percentage of on-time departures using a 95% confidence interval.

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