Editing Z-test (section)

== Example ==

Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively. Our interest is in the scores of 55 students in a particular school who received a mean score of 96. We can ask whether this mean score is significantly lower than the regional mean—that is, are the students in this school comparable to a simple random sample of 55 students from the region as a whole, or are their scores surprisingly low?

First calculate the [[standard error (statistics)|standard error]] of the mean:

:<math>\mathrm{SE} = \frac{\sigma}{\sqrt n} = \frac{12}{\sqrt{55}} = \frac{12}{7.42} = 1.62 </math>

where <math>{\sigma}</math> is the population standard deviation.

Next calculate the [[standard score|''z''-score]], which is the distance from the sample mean to the population mean in units of the standard error:

:<math>z = \frac{M - \mu}{\mathrm{SE}} = \frac{96 - 100}{1.62} = -2.47 </math>

In this example, we treat the population mean and variance as known, which would be appropriate if all students in the region were tested. When population parameters are unknown, a [[Student's t-test|Student's ''t''-test]] should be conducted instead.

The classroom mean score is 96, which is −2.47 standard error units from the population mean of 100. Looking up the ''z''-score in a table of the standard [[normal distribution]] cumulative probability, we find that the probability of observing a standard normal value below −2.47 is approximately 0.5 − 0.4932 = 0.0068. This is the [[one-tailed|one-sided]] [[p-value|''p''-value]] for the null hypothesis that the 55 students are comparable to a simple random sample from the population of all test-takers. The two-sided ''p''-value is approximately 0.014 (twice the one-sided ''p''-value).

Another way of stating things is that with probability 1&nbsp;−&nbsp;0.014&nbsp;=&nbsp;0.986, a simple random sample of 55 students would have a mean test score within 4 units of the population mean. We could also say that with 98.6% confidence we reject the [[null hypothesis]] that the 55 test takers are comparable to a simple random sample from the population of test-takers.

The ''Z''-test tells us that the 55 students of interest have an unusually low mean test score compared to most simple random samples of similar size from the population of test-takers. A deficiency of this analysis is that it does not consider whether the [[effect size]] of 4 points is meaningful. If instead of a classroom, we considered a subregion containing 900 students whose mean score was 99, nearly the same ''z''-score and ''p''-value would be observed. This shows that if the sample size is large enough, very small differences from the null value can be highly statistically significant. See [[statistical hypothesis testing]] for further discussion of this issue.