© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 8. Parameter Estimation Using Confidence Intervals

© 2008 McGraw-Hill Higher Education Confidence Intervals (CI) A range of possible values of a parameter expressed with a specific degree of confidence Confidence interval = point estimate ± error term

© 2008 McGraw-Hill Higher Education With a Confidence Interval (CI): We take a point estimate and use knowledge about sampling distributions to project an interval of error around it A CI provides an interval estimate of an unknown population parameter and precisely expresses the confidence we have that the parameter falls within that interval Answers the question: What is the value of a population parameter, give or take a little known sampling error?

© 2008 McGraw-Hill Higher Education The Level of Confidence The level of confidence is a calculated degree of confidence that a statistical procedure conducted with sample data will produce a correct result for the sampled population

© 2008 McGraw-Hill Higher Education The Level of Significance (α) The level of significance is the difference between the stated level of confidence and “perfect confidence” of 100% This is also called the level of expected error The Greek letter alpha (α) is used to symbolize the level of significance

© 2008 McGraw-Hill Higher Education Confidence and Significance The level of confidence and the level of significance are inversely related – as one increases, the other decreases The level of confidence plus the level of significance sum to 100%. E.g., a level of confidence of 95% has a level of significance of 5%, or a proportion of.05

© 2008 McGraw-Hill Higher Education The Critical Z-score We choose a desired level of confidence by selecting a critical Z-score from the normal distribution table This critical score fits the normal curve and isolates the area of the level of confidence and significance Use the symbol, Z α, for critical scores

© 2008 McGraw-Hill Higher Education Commonly Used Critical Z-scores For a 95% CI of the mean, when n > 121, the critical Z-score = 1.96 SE For a 99% CI of the mean, when n > 121, the critical Z-score = 2.58 SE For a CI of the mean, when n < 121, the critical value is found in a t- distribution table with df = n – 1 (See Chapter 10.)

© 2008 McGraw-Hill Higher Education Steps for Computing Confidence Intervals Step 1. State the research question; draw a conceptual diagram depicting givens (e.g., Figure 8-1 in the text); Step 2. Compute the standard error and the error term Step 3. Compute the LCL and UCL of the CI Step 4. Provide an interpretation in everyday language Step 5. Provide a statistical interpretation

© 2008 McGraw-Hill Higher Education When to Calculate a CI of a Population Mean The research question calls for estimating the population parameter μ X The variable of interest (X) is of interval/ratio level There is a single representative sample from one population

© 2008 McGraw-Hill Higher Education The Error Term The error term of the CI is calculated by multiplying a standard error by a critical Z-score For a CI of the mean, the standard error is the standard deviation divided by the square root of n

© 2008 McGraw-Hill Higher Education Upper and Lower Confidence Limits The upper confidence limit (UCL) provides an estimate of the highest value we think the parameter could have The lower confidence limit (LCL) provides an estimate of the lowest value we think the parameter could have

© 2008 McGraw-Hill Higher Education Calculating the Confidence Limits UCL = sample mean + the error term LCL = sample mean – the error term

© 2008 McGraw-Hill Higher Education Interpretation in Everyday Language Without technical language, this is a statement of the findings for a public audience We state that we are confident to a certain degree (e.g., 95%) that the population parameter falls between the limits of our confidence interval

© 2008 McGraw-Hill Higher Education The Statistical Interpretation The statistical interpretation illustrates the notion of "confidence in the procedure" used to calculate the confidence interval E.g., for the 95% level of confidence we state: If the same sampling and statistical procedures are conducted 100 times, 95 times the true population parameter will be encompassed in the computed intervals and 5 times it will not. Thus, I have 95% confidence that this single CI I computed includes the true parameter

© 2008 McGraw-Hill Higher Education Some Things to Note About a CI of the Mean Typically, the sample standard deviation is used to estimate the standard error (SE) The error term = SE times Z α. A large error term results when either SE or Z α is large The interval reported is an estimate of the population mean, not an estimate of the range of X-scores

© 2008 McGraw-Hill Higher Education Level of Confidence and Degree of Precision The greater the stated level of confidence, the less precise the confidence interval The larger the sample size, the more precise the confidence interval To obtain a high degree of precision and a high level of confidence a researcher must use a sufficiently large sample

© 2008 McGraw-Hill Higher Education Confidence Interval of a Population Proportion With a nominal/ordinal variable, a confidence interval provides an estimate within a range of error of the proportion of a population that falls in the “success” category of the variable

© 2008 McGraw-Hill Higher Education When to Calculate a CI of a Population Proportion We are to provide an interval estimate of the value of a population parameter, P µ, where P = p [of the success category] of a nominal/ordinal variable There is a single representative sample from one population The sample size is sufficiently large that (p smaller ) (n) > 5, resulting in a sampling distribution that is approximately normal

© 2008 McGraw-Hill Higher Education Choosing a Sample Size To obtain a high degree of precision and a high level of confidence a researcher must use a sufficiently large sample Sample size can be chosen to fit a desired level of confidence and range of error The formula for choosing n involves solving for n in the error term of the confidence interval equation

© 2008 McGraw-Hill Higher Education Statistical Follies Scrutinize reports of survey and poll results. Even a major news network may misreport results Often confusion centers around the error term It is plus and minus the error term