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Chapter 414
One-Sample Z-Tests
Introduction
The one-sample z-test is used to test whether the mean of a population is greater than, less than, or not equal
to a specific value. Because the standard normal distribution is used to calculate critical values for the test, this
test is often called the one-sample z-test. The z-test assumes that the population standard deviation is known.
Other PASS Procedures for Testing One Mean or Median
Procedures in PASS are primarily built upon the testing methods, test statistic, and test assumptions that
will be used when the analysis of the data is performed. You should check to identify that the test procedure
described below in the Test Procedure section matches your intended procedure. If your assumptions or
testing method are different, you may wish to use one of the other one-sample procedures available in
PASS–the One-Sample T-Tests and the nonparametric Wilcoxon Signed-Rank Test procedures. The methods,
statistics, and assumptions for those procedures are described in the associated chapters.
If you wish to show that the mean of a population is larger (or smaller) than a reference value by a specified
amount, you should use one of the clinical superiority procedures for comparing means. Non-inferiority,
equivalence, and confidence interval procedures are also available.
The Statistical Hypotheses
In the usual z-test setting, the null (
) and alternative (
) hypotheses for two-sided tests are defined as
: =
versus
:
Rejecting
implies that the mean is not equal to the value
. The hypotheses for one-sided upper-tail
tests are
:
versus
: >
Rejecting
implies that the mean is larger than the value
. This test is called an upper-tail test because
is rejected in samples in which the sample mean is larger than
.
The lower-tail test is
:
versus
: <
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It will be convenient to adopt the following specialize notation for the discussion of these tests.
Parameter PASS Input/Output Interpretation
Population mean. If the data are paired differences, this is the mean
of those differences. This parameter will be estimated by the study.
1 Actual population mean at which power is calculated. This is the
assumed population mean used in all calculations.
0 Reference value. Usually, this is the mean of a reference population. If
the data are paired differences, this is the hypothesized value of the
mean difference.
Population difference. This is the value of
, the difference
between the population mean and the reference value. This
parameter will be estimated by the study.
1 Actual difference at which power is calculated. This is the value of
, the assumed difference between the mean and the
reference value for power calculations.
Assumptions for One-Sample Tests
This section describes the assumptions that are made when you use one of the one-sample tests. The key
assumption relates to normality or non-normality of the data. One of the reasons for the popularity of the t-
test is its robustness in the face of assumption violation. However, if an assumption is not met even
approximately, the significance levels and the power of the t-test are invalidated. Unfortunately, in practice it
often happens that several assumptions are not met. This makes matters even worse! Hence, take the steps
to check the assumptions before you make important decisions based on these tests.
One-Sample Z-Test Assumptions
The assumptions of the one-sample z-test are:
1. The data are continuous (not discrete).
2. The data follow the normal probability distribution.
3. The sample is a simple random sample from its population. Each individual in the population has an
equal probability of being selected in the sample.
4. The population standard deviation is known.
One-Sample T-Test Assumptions
The assumptions of the one-sample t-test are:
1. The data are continuous (not discrete).
2. The data follow the normal probability distribution.
3. The sample is a simple random sample from its population. Each individual in the population has an
equal probability of being selected in the sample.
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Paired T-Test Assumptions
The assumptions of the paired t-test are:
1. The data are continuous (not discrete).
2. The data, i.e., the differences for the matched pairs, follow a normal probability distribution.
3. The sample of pairs is a simple random sample from its population. Each individual in the
population has an equal probability of being selected in the sample.
Wilcoxon Signed-Rank Test Assumptions
The assumptions of the Wilcoxon signed-rank test are as follows (note that the difference is between a data
value and the hypothesized median or between the two data values of a pair):
1. The differences are continuous (not discrete).
2. The distribution of each difference is symmetric.
3. The differences are mutually independent.
4. The differences all have the same median.
5. The measurement scale is at least interval.
Limitations
There are few limitations when using these tests. Sample sizes may range from a few to several hundred. If
your data are discrete with at least five unique values, you can often ignore the continuous variable
assumption. Perhaps the greatest restriction is that your data come from a random sample of the
population. If you do not have a random sample, your significance levels will probably be incorrect.
One-Sample Z-Test Statistic
The one-sample z-test assumes that the data are a simple random sample from a population of normally
distributed values that all have the same mean and variance (known). This assumption implies that the data
are continuous, and their distribution is symmetric. The calculation of the z-test proceeds as follows
=
where
=
and is the value of the mean hypothesized by the null hypothesis that incorporates both
and
.
The significance of the test statistic is determined by computing the p-value. If this p-value is less than a
specified level (usually 0.05), the hypothesis is rejected. Otherwise, no conclusion can be reached.
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Population Size
This is the number of subjects in the population. Usually, you assume that samples are drawn from a very
large (infinite) population. Occasionally, however, situations arise in which the population of interest is of
limited size. In these cases, appropriate adjustments must be made.
When a finite population size is specified, the standard deviation is reduced according to the formula:
= 1
where n is the sample size, N is the population size, is the original standard deviation, and
is the new
standard deviation.
The quantity n/N is often called the sampling fraction. The quantity
1
is called the finite population
correction factor.
Power Calculation for the One-Sample Z-Test
When the standard deviation is known, the power is calculated as follows for a directional alternative (one-
tailed test) in which
>
.
1. Find
such that 1
(
)
=
, where
(
)
is the area to the left of x under the standardized
normal curve.
2. Calculate:
=
+
.
3. Calculate:
=
.
4. Power =
1
(
)
.
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Example 1 – Power after a Study
This example will cover the situation in which you are calculating the power of a z-test on data that have
already been collected and analyzed. For example, you might be playing the role of a reviewer, looking at
the power of a z-test from a study you are reviewing. In this case, you would not vary the means or sample
size since they are given by the experiment. Instead, you investigate the power of the significance tests. You
might look at the impact of different alpha values on the power.
Suppose an experiment involving 100 individuals yields the following summary statistics:
Hypothesized mean (μ0) 100.0
Sample mean (μ1) 110.0
Sample size 100
Given the above data, analyze the power of a z-test which tests the hypothesis that the population mean is
100 versus the alternative hypothesis that the population mean is 110. Consider the power at significance
levels 0.01, 0.05, 0.10 and sample sizes 20 to 120 by 20. The standard deviation is known to be 40.
Note that we have set μ1 equal to the sample mean. In this case, we are studying the power of the z-test for
a mean difference the size of that found in the experimental data.
Setup
If the procedure window is not already open, use the PASS Home window to open it. The parameters for this
example are listed below and are stored in the Example 1 settings file. To load these settings to the
procedure window, click Open Example Settings File in the Help Center or File menu.
Design Tab
_____________ _______________________________________
Solve For ....................................................... Power
Alternative Hypothesis ................................... Two-Sided (H1: μ ≠ μ0)
Population Size .............................................. Infinite
Alpha.............................................................. 0.01 0.05 0.10
N (Sample Size) ............................................. 20 to 120 by 20
μ0 (Null or Baseline Mean) ............................ 100
μ1 (Actual Mean) ........................................... 110
σ (Standard Deviation) ................................... 40
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Output
Click the Calculate button to perform the calculations and generate the following output.
Numeric Reports
Numeric Results
─────────────────────────────────────────────────────────────────────────
Solve For: Power
Hypotheses: H0: μ = μ0 vs. H1: μ ≠ μ0
─────────────────────────────────────────────────────────────────────────
Mean
Sample ──────────────────── Standard
Size Null Actual Difference Deviation Effect
Power N μ0 μ1 μ1 - μ0 σ Size Alpha
──────────────────────────────────────────────────────────────────────────────────────────────────────────────
0.07256 20 100 110 10 40 0.25 0.01
0.15996 40 100 110 10 40 0.25 0.01
0.26130 60 100 110 10 40 0.25 0.01
0.36702 80 100 110 10 40 0.25 0.01
0.46978 100 100 110 10 40 0.25 0.01
0.56466 120 100 110 10 40 0.25 0.01
0.20096 20 100 110 10 40 0.25 0.05
0.35261 40 100 110 10 40 0.25 0.05
0.49069 60 100 110 10 40 0.25 0.05
0.60878 80 100 110 10 40 0.25 0.05
0.70542 100 100 110 10 40 0.25 0.05
0.78191 120 100 110 10 40 0.25 0.05
0.30202 20 100 110 10 40 0.25 0.10
0.47523 40 100 110 10 40 0.25 0.10
0.61489 60 100 110 10 40 0.25 0.10
0.72286 80 100 110 10 40 0.25 0.10
0.80378 100 100 110 10 40 0.25 0.10
0.86298 120 100 110 10 40 0.25 0.10
─────────────────────────────────────────────────────────────────────────
Power The probability of rejecting a false null hypothesis when the alternative hypothesis is true.
N The size of the sample drawn from the population.
μ0 The value of the population mean under the null hypothesis.
μ1 The actual value of the population mean at which power and sample size are calculated.
μ1 - μ0 The difference between the actual and null means.
σ The standard deviation of the population. It measures the variability in the population.
Effect Size The relative magnitude of the effect. Effect Size = |μ1 - μ0|/σ.
Alpha The probability of rejecting a true null hypothesis.
Summary Statements
─────────────────────────────────────────────────────────────────────────
A single-group design will be used to test whether the mean is different from 100 (H0: μ = 100 versus H1: μ ≠ 100).
The comparison will be made using a two-sided, one-sample Z-test, with a Type I error rate (α) of 0.01. The
(known) standard deviation is assumed to be 40. To detect a mean of 110 (corresponding to a difference of 10)
with a sample size of 20, the power is 0.07256.
─────────────────────────────────────────────────────────────────────────
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Dropout-Inflated Sample Size
─────────────────────────────────────────────────────────────────────────
Dropout-
Inflated Expected
Enrollment Number of
Sample Size Sample Size Dropouts
Dropout Rate N N' D
─────────────────────────────────────────────────────────────────────────────
20% 20 25 5
20% 40 50 10
20% 60 75 15
20% 80 100 20
20% 100 125 25
20% 120 150 30
─────────────────────────────────────────────────────────────────────────
Dropout Rate The percentage of subjects (or items) that are expected to be lost at random during the course of the study
and for whom no response data will be collected (i.e., will be treated as "missing"). Abbreviated as DR.
N The evaluable sample size at which power is computed (as entered by the user). If N subjects are evaluated
out of the N' subjects that are enrolled in the study, the design will achieve the stated power.
N' The total number of subjects that should be enrolled in the study in order to obtain N evaluable subjects,
based on the assumed dropout rate. N' is calculated by inflating N using the formula N' = N / (1 - DR), with
N' always rounded up. (See Julious, S.A. (2010) pages 52-53, or Chow, S.C., Shao, J., Wang, H., and
Lokhnygina, Y. (2018) pages 32-33.)
D The expected number of dropouts. D = N' - N.
Dropout Summary Statements
─────────────────────────────────────────────────────────────────────────
Anticipating a 20% dropout rate, 25 subjects should be enrolled to obtain a final sample size of 20 subjects.
─────────────────────────────────────────────────────────────────────────
References
─────────────────────────────────────────────────────────────────────────
Chow, S.C., Shao, J., Wang, H., and Lokhnygina, Y. 2018. Sample Size Calculations in Clinical Research, Third
Edition. Taylor & Francis/CRC. Boca Raton, Florida.
Machin, D., Campbell, M., Fayers, P., and Pinol, A. 1997. Sample Size Tables for Clinical Studies, 2nd Edition.
Blackwell Science. Malden, MA.
Zar, Jerrold H. 1984. Biostatistical Analysis (Second Edition). Prentice-Hall. Englewood Cliffs, New Jersey.
─────────────────────────────────────────────────────────────────────────
This report shows the values of each of the parameters, one scenario per row. The values of power and beta
were calculated from the other parameters.
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Example 2 – Finding the Sample Size
This example will consider the situation in which you are planning a study that will use the one-sample z-test
and want to determine an appropriate sample size. This example is more subjective than the first because
you now have to obtain estimates of all the parameters. In the first example, these estimates were provided
by the data.
In studying deaths from SIDS (Sudden Infant Death Syndrome), one hypothesis put forward is that infants
dying of SIDS weigh less than normal at birth. Suppose the average birth weight of infants is 3300 grams
with a known standard deviation of 663 grams. Use an alpha of 0.05 and power of both 0.80 and 0.90. How
large a sample of SIDS infants will be needed to detect a drop in average weight of 25%? Of 10%? Of 5%?
Note that applying these percentages to the average weight of 3300 yields 2475, 2970, and 3135.
Although a one-sided hypothesis is being considered, sample size estimates will assume a two-sided
alternative to keep the research design in line with other studies.
Setup
If the procedure window is not already open, use the PASS Home window to open it. The parameters for this
example are listed below and are stored in the Example 2 settings file. To load these settings to the
procedure window, click Open Example Settings File in the Help Center or File menu.
Design Tab
_____________ _______________________________________
Solve For ....................................................... Sample Size
Alternative Hypothesis ................................... Two-Sided (H1: μ ≠ μ0)
Population Size .............................................. Infinite
Power............................................................. 0.80 0.90
Alpha.............................................................. 0.05
μ0 (Null or Baseline Mean) ............................ 3300
μ1 (Actual Mean) ........................................... 2475 2970 3135
σ (Standard Deviation) ................................... 663
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Output
Click the Calculate button to perform the calculations and generate the following output.
Numeric Results
─────────────────────────────────────────────────────────────────────────
Solve For: Sample Size
Hypotheses: H0: μ = μ0 vs. H1: μ ≠ μ0
─────────────────────────────────────────────────────────────────────────
Mean
Sample ───────────────────── Standard
Size Null Actual Difference Deviation Effect
Power N μ0 μ1 μ1 - μ0 σ Size Alpha
───────────────────────────────────────────────────────────────────────────────────────────────────────────────
0.86171 6 3300 2475 -825 663 1.244 0.05
0.90861 7 3300 2475 -825 663 1.244 0.05
0.80391 32 3300 2970 -330 663 0.498 0.05
0.90387 43 3300 2970 -330 663 0.498 0.05
0.80085 127 3300 3135 -165 663 0.249 0.05
0.90058 170 3300 3135 -165 663 0.249 0.05
─────────────────────────────────────────────────────────────────────────
This report shows the values of each of the parameters, one scenario per row. Since there were three values
of μ1 and two values of power, there are a total of six rows in the report.
We were solving for the sample size, N. Notice that the increase in sample size seems to be most directly
related to the difference between the two means. The difference in beta values does not seem to be as
influential, especially at the smaller sample sizes.
Note that even though we set the power values at 0.8 and 0.9, these are not the power values that were
achieved. This happens because N can only take on integer values. The program selects the first value of N
that gives at least the values of alpha and power that were desired.
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Example 3 – Finding the Minimum Detectable Difference
This example will consider the situation in which you want to determine how small of a difference between
the two means can be detected by the z-test with specified values of the other parameters.
Continuing with the previous example, suppose about 50 SIDS deaths occur in a particular area per year.
Using 50 as the sample size, 0.05 as alpha, and 0.80 as power, how large of a difference between the means
is detectable?
Setup
If the procedure window is not already open, use the PASS Home window to open it. The parameters for this
example are listed below and are stored in the Example 3 settings file. To load these settings to the
procedure window, click Open Example Settings File in the Help Center or File menu.
Design Tab
_____________ _______________________________________
Solve For ....................................................... μ1 (Search < μ0)
Alternative Hypothesis ................................... Two-Sided (H1: μ ≠ μ0)
Population Size .............................................. Infinite
Power............................................................. 0.80
Alpha.............................................................. 0.05
N (Sample Size) ............................................. 50
μ0 (Null or Baseline Mean) ............................ 3300
σ (Standard Deviation) ................................... 663
Output
Click the Calculate button to perform the calculations and generate the following output.
Numeric Results
─────────────────────────────────────────────────────────────────────────
Solve For: μ1 (Search < μ0)
Hypotheses: H0: μ = μ0 vs. H1: μ ≠ μ0
─────────────────────────────────────────────────────────────────────────
Mean
Sample ───────────────────── Standard
Size Null Actual Difference Deviation Effect
Power N μ0 μ1 μ1 - μ0 σ Size Alpha
─────────────────────────────────────────────────────────────────────────────────────────────────────────────
0.8 50 3300 3037.3 -262.7 663 0.396 0.05
─────────────────────────────────────────────────────────────────────────
With a sample of 50, a difference of 3037.3 - 3300 = -262.7 would be detectable.
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Example 4 – Validation using Chow, Shao, Wang, and
Lokhnygina (2018)
Chow, Shao, Wang, and Lokhnygina (2018) presents an example on pages 45 and 46 of a two-sided one-
sample z-test sample size calculation in which μ0 = 1.5, μ1 = 2.0, σ = 1.0, alpha = 0.05, and power = 0.80.
They obtain a sample size of 32.
Setup
If the procedure window is not already open, use the PASS Home window to open it. The parameters for this
example are listed below and are stored in the Example 4 settings file. To load these settings to the
procedure window, click Open Example Settings File in the Help Center or File menu.
Design Tab
_____________ _______________________________________
Solve For ....................................................... Sample Size
Alternative Hypothesis ................................... Two-Sided (H1: μ ≠ μ0)
Population Size .............................................. Infinite
Power............................................................. 0.80
Alpha.............................................................. 0.05
μ0 (Null or Baseline Mean) ............................ 1.5
μ1 (Actual Mean) ........................................... 2
σ (Standard Deviation) ................................... 1
Output
Click the Calculate button to perform the calculations and generate the following output.
Numeric Results
─────────────────────────────────────────────────────────────────────────
Solve For: Sample Size
Hypotheses: H0: μ = μ0 vs. H1: μ ≠ μ0
─────────────────────────────────────────────────────────────────────────
Mean
Sample ──────────────────── Standard
Size Null Actual Difference Deviation Effect
Power N μ0 μ1 μ1 - μ0 σ Size Alpha
──────────────────────────────────────────────────────────────────────────────────────────────────────────────
0.80743 32 1.5 2 0.5 1 0.5 0.05
─────────────────────────────────────────────────────────────────────────
The sample size of 32 matches Chow, Shao, Wang, and Lokhnygina (2018) exactly.
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Example 5 – Validation using Machin (1997)
Machin, Campbell, Fayers, and Pinol (1997) page 37 presents an example of a one-sample t-test in which μ0
= 0.0, μ1 = 0.2, σ = 1.0, alpha = 0.05, and beta = 0.20. They obtain a sample size of 199. The z-test should give
a similar but slightly lower result because the normal distribution approximates the t distribution very well
at this sample size.
Setup
If the procedure window is not already open, use the PASS Home window to open it. The parameters for this
example are listed below and are stored in the Example 5 settings file. To load these settings to the
procedure window, click Open Example Settings File in the Help Center or File menu.
Design Tab
_____________ _______________________________________
Solve For ....................................................... Sample Size
Alternative Hypothesis ................................... Two-Sided (H1: μ ≠ μ0)
Population Size .............................................. Infinite
Power............................................................. 0.80
Alpha.............................................................. 0.05
μ0 (Null or Baseline Mean) ............................ 0
μ1 (Actual Mean) ........................................... 0.2
σ (Standard Deviation) ................................... 1
Output
Click the Calculate button to perform the calculations and generate the following output.
Numeric Results
─────────────────────────────────────────────────────────────────────────
Solve For: Sample Size
Hypotheses: H0: μ = μ0 vs. H1: μ ≠ μ0
─────────────────────────────────────────────────────────────────────────
Mean
Sample ──────────────────── Standard
Size Null Actual Difference Deviation Effect
Power N μ0 μ1 μ1 - μ0 σ Size Alpha
──────────────────────────────────────────────────────────────────────────────────────────────────────────────
0.80155 197 0 0.2 0.2 1 0.2 0.05
─────────────────────────────────────────────────────────────────────────
The sample size of 197 is very close to and just less than Machin’s result for the t-test.
