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) –interval estimation (we shall calculate an interval about which we are confident the parameter falls within)

Convert deductive to inductive Deductive: P(x-bar L  x-bar  x-bar U )=0.95 Inductive: P(x-bar - z*σ x-bar  μ  x-bar + z*σ x-bar )=0.95

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!!!!!!!

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).

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.

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.

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.

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!)

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.

Sample size for the mean n = z 2 σ 2 /e 2 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.

Sample size for a Proportion n = z 2 p(1-p)/e 2 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.