16. Non-parametric Tests

# 16.2 Median Sign Test

The median sign test is a test of a null hypothesis about the *median*, MD, of a population based on the binomial distribution. To use the test, every subject is assigned a score of , or depending on whether their data point value is greater than, the same as or less than the median.

Since the test is based on the binomial distribution, there are two cases we need to consider. One for small samples and one for large samples where the approximation to the binomial distribution can be used.

**Case 1 : small samples ()**. Here the test statistic is

and the critical statistic, comes from the Sign Test Critical Values Table for a given , 1 or 2 tailed test and a value for where

Reject if .

**Case 2 : large samples ()**. With as defined for case 1, the test statistic is

where

the critical statistic is obtained in the usually way using either the Standard Normal Distribution Table or (recommended) the *t* Distribution Table. Reject if is in the critical region.

There are 3 sets of hypotheses about the null hypothesis median, :

Two-tailed | Left-tailed | Right-tailed |

**Example 16.3** (Small sample size, case 1.).

Given the following snow cone sales data :

18 43 40 16 22

30 29 32 37 36

39 34 39 45 28

36 40 34 39 52

test the conjecture that the median snow cone sales is 40.

*Solution*.

0. Data reduction.

Reduce the data to , and signs relative to :

so (no. of ) = 3, (no. of ) = 15 and .

1. Hypothesis.

2. Critical statistic.

Using the Sign Test Critical Values Table with and for a two-tailed test find

3. Test statistic.

4. Decision.

so reject .

5. Interpretation.

There is enough evidence to reject the claim that the median number of snow cone sales is 40.

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**Example 16.4 : **(Large sample size, case 2.)

We wish to test the claim that the median lifetime of manufactured rubber washers is greater than or equal to 8 years. We are given the following data from a sample of 50 washers :

- 21 washers in our sample last more than 8 years
- 29 washers in our sample last less than 8 years
- (none last exactly 8 years).

*Solution*.

0. Data reduction.

Label the washer that last longer than 8 years with a and the others with a . So (no. of ) = 21 and (no. of ) = 29.

1. Hypothesis

: MD 8 (claim)

: MD 8

2. Critical statistic.

Using the *t* Distribution Table, last line, for a one-tailed test we find

3. Test statistic.

so

4. Decision.

Do not reject .

5. Interpretation.

There is not enough evidence, at , to say that washers last less that 8 years.

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