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?
of M is known to have
a mean of µ and a standard
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
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
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.