Means: Difference from constant (one-sample case)

The one-sample t test is used to determine whether the population mean μ equals some specified value μ₀. A random sample of size N is drawn from a normally distributed population. The true standard deviation in the population is unknown and must be estimated from the data. The null and alternative hypothesis of the t test state:

H₀ : μ − μ₀ = 0
H₁ : μ − μ₀ ≠ 0.

The two-sided ("two tailed") test should be used if there is no restriction on the sign of the deviation from μ₀ assumed in the alternative hypothesis. Otherwise you should use the one-sided ("one tailed") test.

Effect size index

The effect size index d is defined as:

d = (μ − μ₀)/σ

where σ denotes the (unknown) standard deviation in the population. Thus, if μ and μ₀ deviate by one standard deviation then d = 1.

Cohen (1969, p. 38) defined the following conventional values for d:

small d = 0.2
medium d = 0.5
large d = 0.8

Pressing the Determine button on the left side of the effect size label opens the effect size drawer. You can use this drawer to calculate d from μ (Mean H1), μ₀ (Mean H0) and the standard deviation σ (SD σ).

Options

This test has no options.

Examples

We want to test the null hypothesis that the population mean is μ₀ = 10 against the alternative hypothesis that μ = 15. The standard deviation in the population is estimated to be σ = 8. We enter these values in the effect size drawer: Mean H0 = 10, Mean H1 = 15, SD σ = 8 to calculate the effect size d = 0.625.

Next we want to know how many subjects it takes to detect the effect of size d = 0.625 given α = β = .05. We are only interested in increases in the mean and thus choose a one-tailed test.

Select

Type of power analysis: A priori

Input

Tail(s): One
Effect size d: 0.625
α err prob: 0.05
Power (1-β err prob): 0.95

Output

Noncentrality parameter δ: 3.423266
Critical t: 1.699127
df: 29
Total sample size: 30
Actual power: 0.955144

The result indicates that we need at least N = 30 subjects to ensure a power > 0.95. The actual power achieved with this N (0.955144) is slightly higher than the requested power.

Cohen (1969, p. 59) calculates the sample size needed in a two-tailed test that the departure from the population mean is at least 10% of the standard deviation, that is d = 0.1, given α = 0.01 and β ≤ 0.1. The input and output values for this analysis are:

Select

Type of power analysis: A priori

Input

Tail(s): Two
Effect size d: 0.1
α err prob: 0.01
Power (1-β err prob): 0.90

Output

Noncentrality parameter δ: 3.862642
Critical t: 2.579131
df: 1491
Total sample size: 1492
Actual power: 0.900169

G*Power calculates N = 1492 as the required sample size which is slightly higher than the value 1490 estimated by Cohen using his tables.

Related tests

Means: Difference between two dependent means (matched pairs)
Means: Difference between two independent means (two groups)

Implementation notes

The H₀ distribution is the central Student t distribution t(N−1, 0). The H₁ distribution is the noncentral Student t distribution t(N−1, δ), with noncentrality parameter δ = d√N.

Validation

The results were checked against the values produced by GPower 2.0.

References

Cohen, J. (1969). Statistical power analysis for the behavioral sciences. New York, NY: Academic Press.