One-Sample T-Test

The One-sample t-test compares the mean of a normally distributed variable with a hypothesized value (true mean). The test uses the standard deviation of the sample s to estimate the population standard deviation . To run t-test for multiple variables at the same time, simply select the variable with hypothesized values instead of entering the hypothesized value.

# Assumptions

The test assumes the population is normally distributed.

In case the population is not normally distributed, the test statistic has different and unknown distribution and, strictly speaking, the t-test cannot be used. However, according to the central limit theorem, if the sample size is large enough, we can use the t-test even if the normal distribution requirement is not met. Unfortunately there is no easy way to determine what N is large enough. In each particular case there is a separate limit that depends on how strongly the distribution differs from the normal distribution. Some sources quote N=30 as large enough, but even this sample size may turn out to be not large enough. Non-parametric tests may work as an alternative in this case.

# How To

Run Statistics→Basic Statistics→One Sample T-Test...

Select one or more variables.

Select a variable with hypothesized values (if you have a distinct hypothesized value for each variable) or enter a hypothesized value as a number (if you have the same hypothesized value for each variable or you are running the t-test for one variable).

Null hypothesis is defined as H0: . Optionally, you can define the alternative hypothesis H1 in the Advanced Options:

o   H1: less than (lower-tailed) ,

o   H1: not equal (two-tailed),

o   H1: greater than (upper-tailed).

Default value is H1: Not equal (two-tailed).

# Results

Report includes descriptive statistics summary and results of the one sample t-test for each variable.

Mean, Mean LCL, Mean UCL, Standard Error (of Mean), Sample Size – see the Descriptive Statistics procedure for more information.

Hypothesized value (Test Constant, H0 Value) - the hypothesized value (fixed estimate).

Difference – difference between the sample mean and the hypothesized value.

Test statistic– the number   defined as

where  is the standard deviation of the sample,   – sample mean,  – sample size.

Test statistic either exactly follows or closely approximates a t-distribution with N-1 degrees of freedom under the null hypothesis.

d.f. - appropriate degrees of freedom for each variable.

p-value (*-tailed) –p-value for the test, corresponding to the selected H1.

H1: Not equal (two-tailed): If the p-value is less than default  (default value 0.05), statistically, the sample mean is significantly different from the hypothesized value.