For this example assume a normal distribution with a population standard deviation (σ) known to be 100. The null hypothesis is that µ = 500. A random sample of 50 subjects is obtained and the mean of their scores (M) is calculated to be 530. The statistic (M) differs from the parameter specified in the null hypothesis (µ) by 30 points. What is the probability of M differing from µ by 30 or more points?

The sampling distribution of M is known to have a mean of µ and a standard error of

.

For the present example, the mean of the sampling distribution is 500 and the standard error is: 100/7.07 = 14.14. The sampling distribution, then, looks like this:

The mean is 500 and each tick mark represents 1 standard deviation (14 points). The shaded area is the portion of the distribution 30 or more points from µ.

Therefore, the proportion of the area that is shaded is the probability of M differing from µ by 30 or more points? This area can be calculated using the methods described in the section "Area under portions of the normal curve." The area below 470 is 30/14.14 = 2.12 standard deviations below the mean. Therefore, the z score is -2.12.

A z table can be used to find that the probability of a z less than or equal to -2.12 is 0.017. Since the normal distribution is symmetric, the area greater than or equal to 530 is also 0.017. Therefore the probability of obtaining a sample mean 30 or more points from µ (M either ≤ 470 or ≥ 530) is 0.017 + 0.017 = .034. So, the probability that M would be as different or more different from 500 than the difference of 30 obtained in the experiment is 0.034. Or, more compactly: p = .034.