# Confidence Intervals

A confidence interval is a range of estimated values of a population mean based on a sample such that one is confident with a certain probability that the interval straddles the true population mean.

KEY TERMS

Central Limit Theorem
—The variance of the distribution of sample means will be equal to the variance of the population divided by the sample size.
Point estimate
—An estimate of a population parameter, such as a mean or standard deviation, based on the corresponding sample parameter.
Prevalence
—The proportion of a study sample or a population that is affected by a condition or disease during a given time interval.
? (Alpha)
—The “region of acceptance” in statistical hypothesis testing that defines the level confidence with which a researcher can infer values or relationships between variables in a population based on observations of a sample drawn from that population.

To assess depression incidence in the general population, psychologists may measure incidence in a random sample of people in local communities. The sample incidence, is a point estimate of the population incidence of depression μ. Each study sample can have a different number of observations, or N. The larger the N, the more reliable or “representative” the? will be as an estimate of μ.

To see how precise this estimate is, we set up an interval of values of? that will straddle the population mean g in a certain percentage, e.g.,? = 95 percent of the study samples. We define a 100(1-?) percent confidence interval; only 5 percent of possible sample means will be close enough to the value of g to be suitable point estimates of μ. Using a table of values of the standard normal distribution, this corresponds to 1.96 standard deviations (?). The larger the sample N, the lower the sample variance (?2/N), and the tighter the confidence interval will be around?. A symmetric confidence interval around the sample mean will be?± 1.96and the confidence interval can be expressed as (? þ 1.96,? - 1.96

A study of the prevalence of depression in a New England county found that 14 percent of a sample of 300 residents developed symptoms that would to a diagnosis of clinical depression in a given year. The sample standard deviation was 2 percent. The 95 percent confidence interval was (17.9 percent, 10.1 percent). This estimate is close to national prevalence estimates published by the U.S. National Institute of Mental Health (NIMH).

## Resources

BOOKS

Cooper, Harris M., ed. APA Handbook of Research Methods in Psychology. Washington, DC: American Psychological Association, 2012.

Madden, Gregory J., ed. APA Handbook of Behavior Analysis. Washington, DC: American Psychological Association, 2013.

WEBSITE

National Institute of Mental Health (NIMH). “Psychiatric Epidemiology: It's Not Just about Counting Anymore.” http://www.nimh.nih.gov/about/director/index.shtml (accessed November 8, 2014).