3.2 Interval Estimate for the Sample Statistic

The sampling distribution of a continuous sample statistic tells us the probability of finding a range of scores for the sample statistic in a random sample. For example, the average weight of candies in a sample bag is a continuous random variable. The sampling distribution tells us the probability of drawing a sample with average candy weight between 2.0 and 3.6 grams. We can use this range as our interval estimate.

Note that we are reasoning from sampling distribution to sample now. This is not what we want to do in actual research, where we want to reason from sample to sampling distribution to population. We get to that in Section 3.5. For now, assume that we know the true sampling distribution.

Remember that the average or expected value of a sampling distribution is equal to the population value if the estimator is unbiased. For example, the mean weight of yellow candies averaged over a very large number of samples is equal to the mean weight of yellow candies in the population. For an interval estimate, we now select the sample statistic values that are closest to the average of the sampling distribution.

Between which boundaries do we find the sample statistic values that are closest to the population value? Of course, we have to specify what we mean by “closest”. Which part of all samples do we want to include? A popular proportion is 95%, so we want to know the boundary values that include 95% of all samples that are closest to the population value. For example, between which boundaries is average candy weight situated for 95% of all samples that are closest to the average candy weight in the population?

Figure 3.1: Within which interval do we find the sample results that are closest to the population value?

Figure 3.1 shows the sampling distribution of average sample candy weight.

Say, for instance, that 95% of all possible samples in the middle of the sampling distribution have an average candy weight ranging from 1.6 to 4.0 grams. The proportion .95 can be interpreted as a probability. Our sampling distribution tells us that we have 95% probability that the average weight of yellow candies lies between 1.6 and 4.0 grams in a random sample that we draw from this population.

We now have boundary values, that is, a range of sample statistic values, and a probability of drawing a sample with a statistic falling within this range. The probability shows our confidence in the estimate. It is called the confidence level of an interval estimate.