Types of t-tests A t-test is a hypothesis test of the mean of one or two normally distributed populations. Several types of t-tests exist for different situations, but they all use a test statistic that follows a tdistribution under the null hypothesis: Test
Purpose
Example
1 sample t-test
Tests whether the mean of a single
Is the mean height of female college students greater
population is equal to a target value
than 5.5 feet?
Tests whether the difference
Does the mean height of female college students
between the means of two
significantly differ from the mean height of male
independent populations is equal to
college students?
2 sample t-test
a target value paired t-test
Tests whether the mean of the
If you measure the weight of male college students
differences between dependent or
before and after each subject takes a weight-loss pill, is
paired observations is equal to a
the mean weight loss significant enough to conclude
target value
that the pill works?
t-test in
Tests whether the values of
Are high school SAT test scores significant predictors
regression output
coefficients in the regression
of college GPA?
equation differ significantly from zero
An important property of the t-test is its robustness against assumptions of population normality. In other words, t-tests are often valid even when the assumption of normality is violated, but only if the distribution is not highly skewed. This property makes them one of the most useful procedures for making inferences about population means. However, with nonnormal and highly skewed distributions, it might be more appropriate to use nonparametric tests.
Why should I use a 1-sample t-test? To perform this test, choose:
Mac: Statistics > 1-Sample Inference > t
PC: STATISTICS > One Sample > t
Use a 1-sample t-test to estimate the mean of a population and compare it to a target or reference value when you do not know the standard deviation of the population. Using this test, you can do the following:
Determine whether the population mean differs from the hypothesized mean that you specify. Calculate a range of values that is likely to include the population mean.
For example, the manager of a pizza business collects a random sample of pizza delivery times. The manager uses the 1-sample t-test to determine whether the mean delivery time is significantly lower than a competitor's advertised delivery time of 30 minutes. The test calculates the difference between your sample mean and the hypothesized mean relative to the variability of your sample. Usually, the larger the difference and the smaller the variability in your sample, the greater the chance that the population mean differs significantly from the hypothesized mean. The 1-sample t-test also works well when the assumption of normality is violated, but only if the underlying distribution is symmetric, unimodal, and continuous. If the values are highly skewed, it might be appropriate to use a nonparametric procedure, such as a 1-sample sign test. For 1-sample t, the hypotheses are: Null hypothesis H0: μ = µ0
The population mean (μ) equals the hypothesized mean (µ 0).
Alternative hypothesis
Choose one: H1: μ ≠ µ0
The population mean (μ) differs from the hypothesized mean (µ 0).
H1: μ > µ0
The population mean (μ) is greater than the hypothesized mean (µ 0).
H1: μ < µ0
The population mean (μ) is less than the hypothesized mean (µ 0).
Why should I use a 2-sample t test? To perform this test, choose:
Mac: Statistics > 2-Sample Inference > t
PC: STATISTICS > Two Samples > t
Use a 2-sample t-test to do the following:
Determine whether the population means of two independen t groups differ. Calculate a range of values that is likely to include the difference between the population means.
For example, you want to determine whether two grain dispensers are dispensing the same amount of grain. 2-Sample t calculates a confidence interval and does a hypothesis test of the difference between two population means when standard deviations are unknown and samples are drawn independently from each other. This procedure is based on the t-distribution, and for small samples it works best if the data were drawn from distributions that are normal or close to normal. You can have increasing confidence in the results as the sample sizes increase. To do a 2-sample t-test, the two populations must be independent; in other words, the observations from the first sample must not have any bearing on the observations from the second sample. For example, test scores of two separate groups of students are independent, but before-and-after measurements on the same group of students are not independent, although both of these examples have two samples. If you cannot support the assumption of sample independence, reconstruct your experiment to use the paired t-test for dependent populations. The 2-sample t-test also works well when the assumption of normality is violated, but only if the underlying distribution is not highly skewed. With nonnormal and highly skewed distributions, it might be more appropriate to use a nonparametric test. For 2-sample t, the hypotheses are: Null hypothesis H0: μ1 – μ2 = δ0
The difference between the population means (μ 1 – μ2) equals the hypothesized difference (δ 0).
Alternative hypothesis
Choose one: H1: μ1 – μ2≠ δ0
The difference between the population means (μ 1 – μ2) does not equal the hypothesized difference (δ 0).
H1: μ1 – μ2> δ0
The difference between the population means (μ 1 – μ2) is greater than the hypothesized difference (δ 0).
H1: μ1 – μ2≠ δ0
The difference between the population means (μ 1 – μ2) does not equal the hypothesized difference (δ 0).
H1: μ1 – μ2< δ0
The difference between the population means (μ 1 – μ2) is less than the hypothesized difference (δ 0).
Why should I use a paired t test? To perform this test, choose:
Mac: Statistics > 2-Sample Inference > Paired t
PC: STATISTICS > Two Samples > Paired t
Use a paired t-test to do the following:
Determine whether the mean of the differences between two paired samples differs from 0.
Calculate a range of values that is likely to include the population mean of the differences.
Use this analysis to:
Determine whether the mean of the differences between two paired samples differs from 0 (or a target value) Calculate a range of values that is likely to include the population mean of the differences
For example, suppose managers at a fitness facility want to determine whether their weight-loss program is effective. Because the "before" and "after" samples measure the same subjects, a paired t-test is the most appropriate analysis. The paired t-test calculates the difference within each before-and-after pair of measurements, determines the mean of these changes, and reports whether this mean of the differences is statistically significant. A paired t-test can be more powerful than a 2-sample t-test because the latter includes additional variation occurring from the independence of the observations. A paired t-test is not subject to this variation because the paired observations are dependent. Also, a paired t-test does not require both samples to have equal variance. Therefore, if you can logically address your research question with a paired design, it may be advantageous to do so, in conjunction with a paired t-test, to get more statistical power. The paired t-test also works well when the assumption of normality is violated, but only if the underlying distribution is symmetric, unimodal, and continuous. If the values are highly skewed, it might be appropriate to use a nonparametric procedure, such as a 1-sample sign test. For paired t, the hypotheses are: Null hypothesis
H0: μd = μ0
The population mean of the differences (μd) equals the hypothesized mean of the differences (μ0).
Alternative hypothesis
Choose one: H1: μd ≠ μ0
The population mean of the differences (μ d) does not equal the hypothesized mean of the differences (μ 0).
H1: μd > μ0
The population mean of the differences (μ d) is greater than the hypothesized mean of the differences (μ 0).
H1: μd < μ0
The population mean of the differences (μ d) is less than the hypothesized mean of the differences (μ 0).
