## Sign Test

The 1 sample sign non parametric hypothesis test was invented by Dr. Arbuthnot a Scottish physician in the year 1710. Sign test is used to test the null hypothesis that the median of a distribution is equal to some hypothesized value k. The test is based on the direction or the data are recorded as plus and minus signs rather than numerical magnitude, hence it is called Sign test.

The sign test can be used to perform following type of tests

- To determine the preference for one product over the other
- Conduct a test for the median of a single population (one sample sign test)
- To perform a test for the median of paired difference using the data from two dependent samples.

## One Sample Sign Test

The one sample sign test simply computes a significance test of a hypothesized median value for a single data set. The 1 sample sign test is a non parametric hypothesis test used to determine whether statistically significant difference exists between the median of a non-normally distributed continuous data set and a standard. This test is basically concerns the median of a continuous population.

The 1 sample sign test is to compare the total number of observations less than (-ve) or greater than (+) the hypothesized value. The 1 sample sign test is similar to the one-sample Wilcoxon signed-rank test, but less powerful than the Wilcoxon signed test.

The one sample sign test is also considered as non parametric version of one sample t test. Similar to one sample t-test, the sign test for a population median can be one-tailed (right or left tailed) or two-tailed distribution based on the hypothesis.

- Left tailed test- H
_{0}:median≥ Hypothesized value k; H_{1}: median <k - Right tailed test- H
_{0}:median≤ Hypothesized value k; H_{1}: median >k - Two tailed test- H
_{0}: median= Hypothesized value k; H_{1}: median ≠k

## Assumptions of the one sample sign test

- Data is non-normally distributed.
- A random sample of independent measurements for a population with unknown median
- The variable of interest is continuous
- 1 sample test handles non-symmetric data set, that means skewed either to the right or the left.

## Procedure to execute One Sample Sign Non Parametric Hypothesis Test

- State the claim of the test and determine the null hypothesis and alternative hypothesis
- Determine the level of significance
- Assign positive and negative signs to the sample data, and determine the sample size (n)- n is the sum of positive and negative signs
- Find critical value
- Compute the test statistic-
- If n≤ 25 (approx), use y. Where y is the smaller number of positive and negative signs
- For larger sample size, if n > 25, use

- Make a decision, the null hypothesis will be rejected if the test statistic is less than or equal to the critical value
- Interpret the decision in the context of the original claim.

## Example of One Sample Sign Test

Bank of America West Palm Beach, FL branch manager indicates that the median number of savings account customers per day is 64. A clerk from the same branch claims that it was more than 64. Clerk collected the number of savings account customers per day data for 10 random days. Can we reject the branch manager’s claim at 0.05 significance level?

- Null Hypothesis H
_{0}: Savings account customer median = 64; - Alternative Hypothesis H
_{1}: Savings account customer median >64

Assign observations less than 64 with – sign and observations above 64 with + sign

Total number of + values =8

Total number of – values =2

Test statistic is minimum of (8,2) =2

Look at at the Binomial table (10, 0.5)

- Note: 10- is the number of trails
- 0.5 – 50% chance more than the median value and 50% change less than the median value

At 0.05 significance level

Since test statistic 2 is in accept region ( H_{0}), hence accept the null hypothesis. So, there is no significance evidence that the savings account customers per day are more than 64.

**Six Sigma Black Belt Certification 1 Sample Sign Test Questions:**

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**Question 1 :** Which of the following statement is correct with respective to one sample sign test?

(A) More powerful than 1 sample Wilcoxon test

(B) Less powerful than 1 sample Wilcoxon test

(C) More powerful than Mann-Whitney U test

(D) All the above

**Question 2 :** Which of the following distribution of test statistics used in 1 sample sign test?

(A) Binomial Distribution

(B) Poisson Distribution

(C) Hypergeometric Distribution

(D) Exponential Distribution