Confidence Intervals for a Sample Mean

Confidence intervals consist of a sample mean the standard error of the mean (SEM) multiplied by a certain factor. Consider the example of adult height. Imagine selecting a sample of 100 adult males and measuring their heights. On the basis of these 100 measurements, the sample mean and the sample SD can be calculated. Then, on the basis of the SD and the size of the sample (N), the SEM can be calculated as shown in Equation 8.1. Imagine the following hypothetical data  

As noted at the beginning of this section, CIs are calculated as the mean plus or minus the SEM multiplied by a certain factor. In the case of the commonly employed 95% Cl, that factor is 2.0. This value of 2.0 derives from the statement in Section 6.8.1 that 95% of the data points in a Normal distribution fall within 2 SDs of the mean. (Note that while the multiplicative factor of 2 in the case of the 95% Cl is directly related to the statement that 95% of the data points in a Normal distribution fall within + 2 SDs of the mean, the SEM, and not the SD, is used for purposes of calculating CIs.) Hence 

The 95% Cl is often incorrectly conceptualized as stating that the real population mean has a 95% chance of being in the range represented by the lower and upper limits of the Cl. In the example used in the previous paragraph, this equates to a statement that the population mean has a 95% chance of lying in the range of 

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