# Zach’s facts

Zach’s facts have been extracted from the book to remind you of the key concepts you and Zach have learned in each chapter.

Zach's Facts 8.1 The standard error

• A sampling distribution is the frequency distribution of sample statistics (e.g., the mean) from the same population.

• The standard error of the mean is the standard deviation of sample means. As such, it is a measure of how representative a sample mean is likely to be of the population mean. A large standard error (relative to the sample mean) means that there is a lot of variability between the means of different samples, and so the sample we have might not be representative of the population. A small standard error indicates that most sample means are similar to the population mean, and so our sample is likely to be an accurate reflection of the population.

• In general, any parameter (not just the mean) has a sampling distribution and a standard error that tells us how representative the sample estimate is of the true value of that parameter in the population.

Zach's Facts 8.2  Confidence intervals

• A confidence interval for a range of scores is constructed such that the population parameter will fall within this range in 95% of samples. For example, if the parameter is the mean and the limits of the confidence interval are 45 and 55, then this means that the true value of the mean in the population is between 45 and 55 if this sample is one of the 95% in which the confidence interval contains the population value.

• The confidence interval is not an interval within which we are 95% confident that the population parameter (value) will fall. In fact we have no way of knowing whether the interval in our particular sample is one of the 95% that contains the population value or one of the 5% that does not.

• There is not a 95% probability that a confidence interval contains the population parameter (value). The probability that a given interval contains the population value is either 1 (if it is from one of the 95% of samples where the interval hits the population value) or 0 (if it is from one of the 5% of samples where the interval misses the population value). It is impossible to know which of these possibilities applies to a given interval.

• The interval is symmetrical around the sample estimate. For example, if the parameter of interest is the mean then the value of the sample mean will fall halfway between the limits of the confidence interval.