Name |
Formula |
Assumptions or notes |
One-sample -test |
|
(Normal population or n large) and σ known.
(z is the distance from the mean in relation to the standard deviation of the mean). For non-normal distributions it is possible to calculate a minimum proportion of a population that falls within k standard deviations for any k (see: Chebyshev's inequality). |
Two-sample z-test |
|
Normal population and independent observations and σ1 and σ2 are known where is the value of under the null hypothesis |
One-sample t-test |
|
(Normal population or n large) and unknown |
Paired t-test |
|
(Normal population of differences or n large) and unknown |
Two-sample pooled t-test, equal variances |
[3]
|
(Normal populations or n1 + n2 > 40) and independent observations and σ1 = σ2 unknown |
Two-sample unpooled t-test, unequal variances (Welch's t-test) |
[3] |
(Normal populations or n1 + n2 > 40) and independent observations and σ1 ≠ σ2 both unknown |
One-proportion z-test |
|
n .p0 > 10 and n (1 − p0) > 10 and it is a SRS (Simple Random Sample), see notes. |
Two-proportion z-test, pooled for |
|
n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations, see notes. |
Two-proportion z-test, unpooled for |
|
n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations, see notes. |
Chi-squared test for variance |
|
df = n-1
• Normal population |
Chi-squared test for goodness of fit |
|
df = k − 1 − # parameters estimated, and one of these must hold.
• All expected counts are at least 5.[4]
• All expected counts are > 1 and no more than 20% of expected counts are less than 5[5] |
Two-sample F test for equality of variances |
|
Normal populations Arrange so and reject H0 for [6] |
Regression t-test of |
|
Reject H0 for [7] *Subtract 1 for intercept; k terms contain independent variables. |
In general, the subscript 0 indicates a value taken from the null hypothesis, H0, which should be used as much as possible in constructing its test statistic. ... Definitions of other symbols:
|
- = sample variance
- = sample 1 standard deviation
- = sample 2 standard deviation
- = t statistic
- = degrees of freedom
- = sample mean of differences
- = hypothesized population mean difference
- = standard deviation of differences
- = Chi-squared statistic
|
- = sample proportion, unless specified otherwise
- = hypothesized population proportion
- = proportion 1
- = proportion 2
- = hypothesized difference in proportion
- = minimum of and
- = F statistic
|
|