# 11.1.2 - One-Sample Wilcoxon

11.1.2 - One-Sample Wilcoxon

In this section, we briefly present the one-sample Wilcoxon test. This test was developed by Frank Wilcoxon in 1945. It is considered one of the first “nonparametric” tests developed.

The hypotheses are the similar to the ones presented previously for the Sign Test:

#### Null

$$H_0\colon \eta=\eta_0$$

#### Alternative

$$H_a\colon \eta>\eta_0$$

$$H_a\colon \eta<\eta_0$$

$$H_a\colon \eta\ne\eta_0$$

The Wilcoxon test needs additional assumptions, however. They are:

1. The random variable of interest is continuous
2. The probability distribution of the population is symmetric.

If we compare the assumptions of the Wilcoxon test to the Sign Test, the Wilcoxon test requires the distribution to be symmetric. For example, we should not be making conclusions for the IRS data using the Wilcoxon test because the data is right-skewed.

The test statistic is typically denoted as $$W$$. We will not go into details on how this statistic is found as it involves ranks.

## One-Sample Wilcoxon Test

Minitab will conduct the one-sample Wilcoxon test.

1. Choose Stat  > Nonparametrics > 1-sample Wilcoxon
2. Enter the 'variable', the 'hypothesized value', and the correct 'alternative'.
3. Choose OK .

## Example 11-3: Checkout Time (Wilcoxon Test)

Fresh N Friendly food store advertises that their checkout waiting times is four minutes or less. An angry customer wants to dispute this claim. He takes a random sample of shoppers at the peak time and records their checkout times. Can he dispute their claim at significance level 10%?

Checkout times:

3.8, 5.3, 3.5, 4.5, 7.2, 5.1

Use Minitab to conduct the 1-sample Wilcoxon Test. Compare the conclusion to the one found using the one-sample t-test. Lesson 6b.4 More Examples

The hypotheses are $$H_0:\eta=4$$ versus $$H_0:\eta>4$$. The sample size is small here and may not even be reasonable to assume that the data is symmetric. We get the following output from Minitab:

#### Method

$$\eta$$: median of time

#### Descriptive Statistics

 Sample N Median time 6 4.8

#### Test

Null hypothesis

Alternative hypothesis

Ho: $$\eta$$ = 4

H1: $$\eta$$ > 4

N for Wilcoxon

Sample N for Test Wilcoxon Statistic P-Value
time 6 17.50 0.086

The p-value for this test is 0.086. The p-value is less than our significance level and therefore we reject the null hypothesis. There is enough evidence in the data to suggest the population median time is greater than 4.

If we assume the data are normal and perform a test for the mean, the p-value was 0.0798.

At the 10% level, the data suggest that both the mean and the median are greater than 4.

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