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Publication date: 04/21/2023

Power for One Sample Mean

Use the Power for One Sample Mean Explorer to determine a sample size for a hypothesis test about one mean. Select DOE > Sample Size Explorers > Power > Power for One Sample Mean. Explore the trade-offs between variability assumptions, sample size, power, significance, and the hypothesized difference to detect. Sample size and power are associated with the following hypothesis test:

Equation shown here

versus the two-sided alternative:

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or versus a one-sided alternative:

Equation shown here or Equation shown here

where μ is the true mean and μ0 is the null mean or reference value. The difference to detect is an amount, Δ, away from μ0 that one considers important to detect. For the same significance level and power, a larger sample size is needed to detect a small difference than to detect a large difference. It is assumed that the population of interest is normally distributed with mean μ and standard deviation σ.

Power Explorer for One Sample Mean Settings

Set study assumptions and explore sample sizes using the radio buttons, text boxes, and menus. The profiler updates as you make changes to the settings. Alternatively, change settings by dragging the cross hairs on the profiler curves.

Test Type

Specifies a one or two-sided hypothesis test.

Preliminary Information

Alpha

Specifies the probability of a type I error, which is the probability of rejecting the null hypothesis when it is true. It is commonly referred to as the significance level of the test. The default alpha level is 0.05.

Population Standard Deviation

Specifies the distribution for calculations.

Yes

Specifies a known standard deviation, calculations use the z distribution.

No

Specifies an unknown standard deviation, calculations use the t distribution.

Power Explorer for One Sample Mean Profiler

The profiler enables you to visualize the impact of sample size assumptions on the power calculations.

Solve for

Enables you to solve for sample size, the difference to detect, or the assumed standard deviation.

Power

Specifies the probability of rejecting the null hypothesis when it is false. With all other parameters fixed, power increases as sample size increases.

Sample Size

Specifies the total number of observations (runs, experimental units, or samples) needed for your experiment.

Difference to Detect

Specifies smallest difference between the true mean and the hypothesized or reference mean that you want to be able to declare statistically significant.

Std Dev (σ)

Specifies the assumed population standard deviation.

Tip: Use a standard deviation of 1 to estimate the sample size needed to detect differences measured in standard deviation units.

Power Explorer for One Sample Mean Options

The Explorer red triangle menu and report buttons provide additional options:

Simulate Data

Opens a data table of simulated data based on the explorer settings. View the simulated response column formula for the settings used.

Make Data Collection Table

Creates a new data table that you can use for data collection. The table includes scripts to facilitate data analysis.

Save Settings

Saves the current settings to the Saved Settings table. This enables you to save a set of alternative study plans. See Saved Settings in the Sample Size Explorers.

Help

Opens JMP help.

Statistical Details for Power Explorer for One Sample Mean

The one sample mean calculations are based on the t test when σ is unknown and estimated from sample data. For the case when σ is known, the calculations use the z test. For the case when σ is unknown, the power is calculated according to the alternative hypothesis.

For a one-sided, higher alternative:

Equation shown here

For a one-sided, lower alternative:

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For a two-sided alternative:

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where:

α is the significance level

n is the sample size

σ is the assumed population standard deviation

δ is the difference to detect

t1-α,ν is the (1 - α)th quantile of the central t-distribution with ν degrees of freedom

T(t; ν, λ) is the cumulative distribution function of the non-central t distribution with ν degrees of freedom and non-centrality parameter λ.

When σ is known the z distribution is used in the above equations for the power calculations. Because closed-form solutions for δ and n do not exist, numerical routines are used to solve for them.

Want more information? Have questions? Get answers in the JMP User Community (community.jmp.com).