Confidence Interval

Quantitative research in educational psychology benefits from using sample data to describe the wider population of interest. A confidence interval (CI) is a statistical approach that uses sample data to determine a range of values on a particular statistic (e.g., mean) for a sample from the population. In theory, a confidence interval is based on an infinite number of random samples from the population. For each of these random samples, a statistic of interest, such as the mean, could be calculated. Of course, it is highly unlikely that each of these means will be equal to one another, and even more unlikely that any or all of the means will equal the population mean. Thus, it is generally safe to assume that error will be present when estimating a population parameter with a statistic based on a random sample. Comparing the CI to the sample mean provides the researcher a means for determining the magnitude of error.

A CI consists of two numbers, often referred to as lower and upper bounds, between which the value of a population parameter is expected to fall. For instance, the CI for a population mean is typically reported in the following format:

where CI PI indicates the probability factor for the respective confidence interval, and μ represents the population mean. In general, probability factors of 90, 95, and 99 are used to construct CIs. For interpretation purposes, these probability factors are converted to percentages. For instance, using a probability factor of 95 results in a 95% CI, which indicates that the researcher can be 95% confident that the population mean falls within the lower and upper bounds of the interval. As another example, consider a95% CI for mean scores on a standardized mathematics assessment for seventh graders at Jayhawk Middle School:

This CI indicates that 95% of random samples taken from seventh graders at Jayhawk Middle School will yield a mean math score greater than 70 and less than 90.

The latter example can also be used to illustrate the role of measurement error in estimating population parameters. Begin by assuming that the CI for seventh graders at Jayhawk Middle School was based on a sample mean that equals the population mean of 80. Furthermore, suppose a random sample of seventh graders from Jayhawk Middle School yielded a mean mathematics assessment score of 81. An additional, independent, random sample taken from the same population of seventh graders at Jayhawk Middle School yielded a mean mathematics assessment score of 77. In both cases, the sample means do not equal the population mean, nor do they equal one another.

The amount of measurement error is used to construct a CI around a particular parameter. This measurement error is referred to as the standard error. The standard error provides a measure of the variability of the sample statistic around the population parameter. For instance, the standard error of the mean is calculated using the following formula:

where σX is the standard error of the mean, s is the standard deviation of the sample, and N is the sample size. The distribution of an infinite number of randomly sampled means is known to be normal and can be converted to a standard normal distribution. Values can be taken from the standard normal distribution to construct CIs. For instance, a 95% CI requires the use of a standard value of 1.96. This value is multiplied by the standard error of the mean in order to convert the CI to the original unit of measurement. Therefore, the equation for constructing a 95% confidence interval

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