© 2008 McGraw-Hill Higher Education

The Statistical Imagination

• Chapter 8. Parameter Estimation Using Confidence Intervals

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Confidence Intervals (CI)

• A range of possible values of a parameter expressed with a specific degree of confidence

• Confidence interval = point estimate ± error term

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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?

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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

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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

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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

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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, Zα, for critical scores

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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.)

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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

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When to Calculate a CI of a Population Mean

• The research question calls for estimating the population parameter μX

• The variable of interest (X) is of interval/ratio level

• There is a single representative sample from one population

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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

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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

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Calculating the Confidence Limits

• UCL = sample mean + the error term

• LCL = sample mean – the error term

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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

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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

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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 Zα . A large error term results when either SE or Zα is large

• The interval reported is an estimate of the population mean, not an estimate of the range of X-scores

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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

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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

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When to Calculate a CI of a Population Proportion

• We are to provide an interval estimate of the value of a population parameter, Pµ , 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 (psmaller) (n) > 5, resulting in a sampling distribution that is approximately normal

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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

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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