Statistical Test

One-Sample t-Test
A one-sample t-test is used to test whether a population mean is significantly different from some hypothesized value.
The one-sample t-test can be used when the population variances are equal or unequal, and with large or small samples.

• Define hypothesis (one-tailed or two-tailed)
• Specify significance level and determine degrees of freedom (n-1)
• Compute test statistic (t-statistic)
• Compute P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic.

Two-Sample t-Test
A two-sample t-test is used to test the difference between two population means. A common application is to determine whether the means are equal. The two-sample t-test can be used when the population variances are equal or unequal, and with large or small samples.

• Define hypothesis (one-tailed or two-tailed)
• Specify significance level and determine degrees of freedom (n-1)
• Compute test statistic (t-statistic)
• Compute P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic.

z-Test

Chi Square Test

ANNOA

This test determines whether the means of three or more groups are different. ANOVA uses F-tests to statistically test the equality of means.

F = variation between group / variation within group

Test Statistics

mean

proportion

difference between means

difference between proportions

z-score - How many standard deviations an element is from the mean

$\begin{aligned} z & = \frac {X - \mu} {\sigma} \qquad (mean) \\ z & = \frac {p - P} {\sigma} \qquad (proportion) \\ \end{aligned}$

• A z-score less than 0 represents an element less than the mean
• A z-score greater than 0 represents an element greater than the mean
• A z-score equal to 0 represents an element equal to the mean
• A z-score equal to 1 represents an element that is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean; etc.
• A z-score equal to -1 represents an element that is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean; etc.
• If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; and about 99% have a z-score between -3 and 3.

t-statistic - Distribution of t-statistic is called the t-distribution or the Student t distribution

$t = \frac {\bar x - \mu} {s / \sqrt{n}}$

chi-square

f-statistic - ratio of two variances