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A one-sample z-test is used to test whether a population parameter is
significantly different from some hypothesized value.

Here is how to use the test.

Define hypotheses.
The table below shows three sets of null and alternative hypotheses.
Each makes a statement about how the true population mean
μ is related to some hypothesized value
M.
(In the table, the symbol ≠ means " not equal to ".)

Set

Null hypothesis

Alternative hypothesis

Number of tails

1

μ = M

μ ≠ M

2

2

μ > M

μ < M

1

3

μ < M

μ > M

1

Specify significance level. Often, researchers choose
significance levels
equal to
0.01, 0.05, or 0.10; but any value between 0 and
1 can be used.

Compute test statistic. The test statistic is a z-score (z) defined by
the following equation.

z = (x
- M )
/ [ σ /sqrt(n) ]

where
x is the observed sample mean,
M is the hypothesized population mean (from the null hypothesis), and
σ is the
standard deviation
of the population.

Compute P-value. The P-value is the probability of observing a
sample statistic as extreme as the test statistic. Since the
test statistic is a z-score, use the
Normal Distribution Calculator
to assess the probability associated with the z-score.

Evaluate null hypothesis. The evaluation involves comparing the P-value to the
significance level,
and rejecting the null hypothesis when the P-value is less than
the significance level.

The one-sample z-test can be used when the population is normally
distributed, and the population variance is known.