# 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.
Donnerstag, 23. 05. 2013