## It’s A Math, Math World (The Sign Test)

So far, every hypothesis test we have considered had required an assumption of normality or near-normality. Sometimes, though, the distribution of the population is non-normal or unknown and we use a *nonparametric test*. An easy example of such a test is the *sign test* for testing the value of a median of a population.

**Example:**

A seed company wants to market a new type of seed that would reportedly produce a greater yield than the old type of seed. Thirteen farmers agree to test grown the new seed in one acre and test grown the old variety on another acre and look at the difference in wheat yield.

**BUSHELS OF WHEAT FROM 2 TYPES OF SEED**

FARM |
NEW VARIETY (y1) |
OLD VARIETY (y2) |
DIFFERENCE D= y1 – y2 |
Sign of D |

1 | 34 | 27 | 7 | + |

2 | 45 | 25 | 20 | + |

3 | 30 | 38 | -8 | – |

4 | 30 | 42 | -12 | – |

5 | 48 | 21 | 27 | + |

6 | 35 | 22 | 13 | + |

7 | 32 | 37 | -5 | – |

8 | 46 | 30 | 16 | + |

9 | 41 | 32 | 9 | + |

10 | 23 | 38 | -15 | – |

11 | 42 | 26 | 16 | + |

12 | 43 | 33 | 10 | + |

13 | 65 | 68 | -3 | – |

We are unsure of the distributions of the two varieties of wheat so we will use the sign test.

y1 = yield from the new variety (in bushels)

y2 = yield from the old variety (in bushels)

D = y1 – y2

**Hypotheses:**

H0: Median(D) = 0

Ha: Median(D) > 0

**Level of significance: α =0.05**

**Test statistics and Observed value:**

We count the number of values of D that are above zero (plus signs) from above table.

X = number of plus signs = 8

**Critical Region**: We perform a right tailed test. Looking at the table of Binomial probabilities, with n=13 and x=8, we see if we use the values 10, 11, 12 and 13 for the critical region then:

Α = P(10) + P(11) + P(12) + P(13) = .035 + .010 + .002 + 0 = .047 which is close to .05, hence the critical region will consist of the following x-values: 10, 11, 12 and 13.

**Decision**: The observed value (x=8) is not in the critical region so we do not reject the null hypothesis and we conclude that there is not enough evidence to conclude that the new variety of seed is more effective than the old variety.

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Ok, these are all observations after the fact. How do you determine for something that’s a bit more dynamic, and needs to be updated constantly? Can you apply the same technique to a poker situation?