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Chapter 8

Chapter 8. Confidence Interval Estimation. 8.1 Confidence Interval Estimation of the Mean . This section deals with the case of known σ. There are two kinds of estimation: point estimation (we prefer to use unbiased estimators)

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Chapter 8

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  1. Chapter 8 Confidence Interval Estimation

  2. 8.1 Confidence Interval Estimation of the Mean • This section deals with the case of known σ. • There are two kinds of estimation: • point estimation (we prefer to use unbiased estimators) • interval estimation (we shall calculate an interval about which we are confident the parameter falls within)

  3. Convert deductive to inductive • Deductive: P(x-barLx-bar  x-barU)=0.95 • Inductive: P(x-bar - z*σx-bar μ  x-bar + z*σx-bar)=0.95

  4. Example Consider Figure 8.1 • You know μ and σ. • You take 5 samples and calculate the interval estimate for each sample. • Not every interval is successful!!!!!!!

  5. Results • Formula for confidence interval (CI) appears on page 263. Note 3 alternative ways to write. Also note 1 - α • Interpretation appears on page 262. • Appropriate conclusion: We are 95% confident that the true mean falls between ___ and ____. • Note tradeoffs between confidence and size of interval (examples 8.1&8.2).

  6. 8.2 CI Estimation of the Mean when σ is unknown. • Assuming that σ is known is very often unrealistic. • When a parameter is unknown, it must be estimated. Use S to estimate σ. • You can get away with this estimation under certain conditions.

  7. Conditions • The R.V. X is assumed to be approximately normally distributed (remember: we didn’t have to make this assumption before!). • We can’t use the normal distribution for x-bar. We have to use the “t” or “Student’s t” distribution.

  8. Student’s t Distribution • Looks a lot like Normal Distribution (Figure 8.4) • Need two items to look up a value of “t” • appropriate degrees of freedom • confidence level / area under curve • The CI for unknown σ appears on page 268.

  9. 8-3: CI Estimation for the Proportion • So, you have a set of categorical data. • Use the sample proportion. • You can estimate the population proportion: • take the sample proportion and adjust it • adjustment = appropriate z * appropriate standard deviation • page 273 (read the fine print on page 274!)

  10. 8-4: Determining Sample Size • You will occasionally want to determine the sample size based on your acceptable sampling error. • You might also possibly change how confident you want to be based on the expense of obtaining samples. • You can determine the required sample size for estimating either the mean or the proportion.

  11. Sample size for the mean • n = z2σ2/e2 • use the confidence level to find z • an expert should decide on “e” --the sampling error--how much sampling error can be tolerated? • σ might come from history, or from an educated guess, or from an independent (pilot) study.

  12. Sample size for a Proportion • n = z2p(1-p)/e2 • use the confidence level to find z • an expert should decide on “e” --the sampling error--how much sampling error can be tolerated? • For p, use either past data or p=0.5.

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