Statistics problems often involve comparisons between two independent sample means. This lesson explains how to compute probabilities associated with differences between means.
Difference Between Means: Theory
Suppose we have two populations with means equal to μ1 and μ2. Suppose further that we take all possible samples of size n1 and n2. And finally, suppose that the following assumptions are valid.
Given these assumptions, we know the following.
It is straightforward to derive the last bullet point, based on material covered in previous lessons. The derivation starts with a recognition that the variance of the difference between independent random variables is equal to the sum of the individual variances. Thus,
σ2d = σ2 (x1 - x2) = σ2 x1 + σ2 x2
If the populations N1 and N2 are both large relative to n1 and n2, respectively, then
σ2 x1 = σ21 / n1
σ2 x2 = σ22 / n2
σd2 = σ12 / n1 + σ22 / n2
σd = sqrt( σ12 / n1 + σ22 / n2 )
Difference Between Means: Sample Problem
In this section, we work through a sample problem to show how to apply the theory presented above. In this example, we will use Stat Trek's Normal Distribution Calculator to compute probabilities.
Normal Distribution Calculator
The normal calculator solves common statistical problems, based on the normal distribution. The calculator computes cumulative probabilities, based on three simple inputs. Simple instructions guide you to an accurate solution, quickly and easily. If anything is unclear, frequently-asked questions and sample problems provide straightforward explanations. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.Normal Distribution Calculator
For boys, the average number of absences in the first grade is 15 with a standard deviation of 7; for girls, the average number of absences is 10 with a standard deviation of 6.
In a nationwide survey, suppose 100 boys and 50 girls are sampled. What is the probability that the male sample will have at most three more days of absences than the female sample?
The correct answer is B. The solution involves three or four steps, depending on whether you work directly with raw scores or z-scores. The "raw score" solution appears below:
Thus, the probability that the difference between samples will be no more than 3 days is 0.035.
Alternatively, we could have worked with z-scores (which have a mean of 0 and a standard deviation of 1). Here's the z-score solution:
Of course, the result is the same, whether you work with raw scores or z-scores.
Note: Some analysts might have used the t-distribution to compute probabilities for this problem. We chose the normal distribution because the population variance was known and the sample size was large; but either choice would have been acceptable. In a previous lesson, we offered some guidelines for choosing between the normal and the t-distribution.