One-Sample t-Test
A t-test is used when
the population variance (σ²) is unknown or
when the sample size is small (n ≤ 30)
It is a parametric test used to evaluate hypotheses concerning a single sample.
The specified value (expected mean) is obtained from a theoretical model, previous research, or
other sources.
The test examines the null hypothesis that the sample mean is equal to a specified population
value.
Assumptions:
Randomly selected sample
Continuous numerical dependent variable
Normal distribution of the dependent variable (the test is robust to some deviation from
normality, and this robustness increases with sample size)
No significant outliers
If the sample is small and the distribution is unknown, non-parametric tests should be used
instead of the t-test or z-test.
Variable Type:
One dependent numerical variable
Test Statistic:
The test statistic follows a t-distribution with a number of degrees of freedom (df = n - 1).
The t-distribution is a modified version of the standard normal distribution, adjusted for:
Small sample sizes
Unknown population standard deviation
Characteristics of the t-distribution:
, Similar to the normal distribution but with heavier tails (greater probability of extreme
values)
Mean is always 0
Standard deviation is 1
Shape of the curve depends on degrees of freedom
For small samples, estimating the population is more difficult, increasing the chance of
extreme outcomes
Symmetrical, bell-shaped curve that is lower and wider than the standard normal curve
As sample size increases, the t-distribution converges toward the normal distribution.
The fewer the degrees of freedom, the heavier the tails, due to greater variability in the
sample.
SPSS Procedure (for One-Sample t-Test):
1. Procedure execution
2. Case Processing Summary
3. Descriptives
4. One-Sample Statistics
5. One-Sample Test
Handling Missing Values in SPSS:
Missing Values – “Exclude cases analysis by analysis”:
This option ensures that only complete cases for a specific variable are included in that specific
analysis, while allowing other analyses to use the same case if data for their variables is
complete.
SPSS Output Interpretation:
One-Sample Test Table shows:
o t-value (t)
o Degrees of freedom (df)
o Two-tailed significance (Sig. (2-tailed))
A larger absolute value of the t-statistic indicates a greater likelihood of a significant
difference between the sample mean and the specified population mean.
A t-test is used when
the population variance (σ²) is unknown or
when the sample size is small (n ≤ 30)
It is a parametric test used to evaluate hypotheses concerning a single sample.
The specified value (expected mean) is obtained from a theoretical model, previous research, or
other sources.
The test examines the null hypothesis that the sample mean is equal to a specified population
value.
Assumptions:
Randomly selected sample
Continuous numerical dependent variable
Normal distribution of the dependent variable (the test is robust to some deviation from
normality, and this robustness increases with sample size)
No significant outliers
If the sample is small and the distribution is unknown, non-parametric tests should be used
instead of the t-test or z-test.
Variable Type:
One dependent numerical variable
Test Statistic:
The test statistic follows a t-distribution with a number of degrees of freedom (df = n - 1).
The t-distribution is a modified version of the standard normal distribution, adjusted for:
Small sample sizes
Unknown population standard deviation
Characteristics of the t-distribution:
, Similar to the normal distribution but with heavier tails (greater probability of extreme
values)
Mean is always 0
Standard deviation is 1
Shape of the curve depends on degrees of freedom
For small samples, estimating the population is more difficult, increasing the chance of
extreme outcomes
Symmetrical, bell-shaped curve that is lower and wider than the standard normal curve
As sample size increases, the t-distribution converges toward the normal distribution.
The fewer the degrees of freedom, the heavier the tails, due to greater variability in the
sample.
SPSS Procedure (for One-Sample t-Test):
1. Procedure execution
2. Case Processing Summary
3. Descriptives
4. One-Sample Statistics
5. One-Sample Test
Handling Missing Values in SPSS:
Missing Values – “Exclude cases analysis by analysis”:
This option ensures that only complete cases for a specific variable are included in that specific
analysis, while allowing other analyses to use the same case if data for their variables is
complete.
SPSS Output Interpretation:
One-Sample Test Table shows:
o t-value (t)
o Degrees of freedom (df)
o Two-tailed significance (Sig. (2-tailed))
A larger absolute value of the t-statistic indicates a greater likelihood of a significant
difference between the sample mean and the specified population mean.
