Confidence Intervals/Hypothesis Testing/Effect Sizes

Confidence Intervals

Using the foregoing example, assume that the correlation (ρ) between the test scores and ratings is .35 in the population. This ρ of .35 is called a parameter because it is based on a population, whereas the r of .25 is called a statistic because it is based on a sample. If another sample of 100 cases were drawn randomly from the same population with an infinite number of cases, the r calculated using test and ratings data from the new sample would probably differ from both the ρ of .35 and the r of .25 calculated for the first sample. We can continue sampling and calculating r an infinite number of times, and each rs would be an estimate of ρ. Typically, r would be distributed around ρ, with some being smaller than .35 and others being larger. If the mean of all possible rs equals ρ, then each r is said to be an unbiased estimate of ρ. The distribution of r is called the sampling distribution of the sample correlation, and its standard deviation is called the standard error (SE). The SE of r can be calculated as

where N is the sample size. Notably, as N increases, SE decreases.

A confidence interval (CI) is the portion of the sampling distribution into which a statistic (e.g., r) will fall a prescribed percentage of the time. For example, a 95% CI means that a statistic will fall between the interval's lower and upper limits 95% of the time. The limits for the 95% CI can be calculated as follows:

If a 99% CI were desired, the value 2.58 would be substituted for 1.96. These two levels of CI are those most commonly used in practice.

Let us compute CI using ρ= .35, N = 100, the common assumption that the distribution of r is normal, and Equation 1, SE = (1 − .352)/√100 = .088. Using Equation 2, the lower limit is .35 − (1.96)(.088) = .178, and the upper limit is .35 + (1.96)(.088) = .472. This example CI can be interpreted as follows: If an infinite number of random samples of 100 each were drawn from the population of interest, 95% of the rs would fall between .178 and .472, and 5% would fall outside the CI.

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