(a) Run Test
The Run Test, also called the Runs Test for Randomness, is a non-parametric test used to determine if a sequence of data points is randomly distributed. It checks for the occurrence of runs, which are sequences of similar items (e.g., consecutive positive or negative numbers) in the data. The test evaluates whether the number of runs in the data is significantly different from what would be expected in a random sequence.
- Null Hypothesis (H₀): The data is randomly ordered.
- Alternative Hypothesis (H₁): The data is not randomly ordered.
A run is defined as a sequence of consecutive identical values (e.g., all positive or all negative signs). If the number of runs is too high or too low compared to the expected number under randomness, the null hypothesis of randomness is rejected. This test is often applied in quality control, finance, or any field where randomness in a sequence needs to be tested.
(b) Sign Test
The Sign Test is a simple non-parametric test used to compare the medians of two related samples or to test whether the median of a single sample is equal to a hypothesized value. It is particularly useful when the data is ordinal or when the assumptions of parametric tests like the t-test are not met.
For two related samples, the test examines the signs (+ or -) of the differences between paired observations, ignoring the magnitude of the differences. For a single sample, the test compares each observation to the hypothesized median.
- Null Hypothesis (H₀): The median of the differences is zero (for paired samples), or the median is equal to the hypothesized value (for a single sample).
- Alternative Hypothesis (H₁): The median of the differences is not zero, or the median differs from the hypothesized value.
The Sign Test is easy to use and requires no assumptions about the underlying distribution, but it is less powerful than other tests like the Wilcoxon signed-rank test.
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