# Levels of Significance, p-Values and Standard Deviations

I have four restaurants with a nearest neighbor analysis defining the restaurant pattern as clustered, random, or dispersed.

Italian: Observed mean distance/expected mean distance = 0.67 Less than 1% likelihood it being clustered is random

Z score = 6.3 standard deviations

Significance Levels: 0.01; 0.05; 0.10; Random; 0.10; 0.05; 0.01

Critical Values: -2.56; -1.96; -1.65; 1.65; 1.96; 2.58

Japanese: Observed Mean distance/expected mean distance = 0.25 Less than 1% it being clustered is random

Z score = -9.4 SD

Sig and critical values are the same

Korean: OMD/EMD = 0.95 Neither clustered nor random; it is right in the center of both

Z score = -0.3 SD

Sig And Critical are the same

French: Less than 1% likelihood that its being dispersed as clustered is random

OMD/EMD = 0.42

A score = -4.9 SD

Sig and Critical are the same.

#### Solution Preview

SIGNIFICANCE LEVEL (also sometimes called P-VALUE):

Suppose you have some random quantity x with some type of average xa (this average is usually taken to be the MEAN, but it can be another popular measure such as MEDIAN, MODE or something fancier still).

When in a particular instance you measure the value of x to be some xe which is different from xa, you can ask (and calculate, if you know the probability distribution) what is the probability of a random result to deviate from the average by as much or more than what happened in this particular instance, that is by more than |xa-xe|.

Let us denote this probability as P(>|xe-xa|) - this notation is a bit cumbersome but informative.

This probability is called SIGNIFICANCE LEVEL or P-VALUE under the following circumstances.

When we want to test some hypothesis, usually called NULL HYPOTHESIS, we make some experiments to verify it. We choose some numerical quantity xa to quantify some prediction by the null hypothesis, but in the testing we get value xe instead. At this stage we ask ourselves how significant is the fact that we got xe rather than xa. Here comes to our help the probability

P(>|xe-xa|) - the p-value. The p-value is the probability that the discrepancy between xa and xe happened accidentally rather than because our hypothesis is wrong.

The smaller it is the ...

#### Solution Summary

Levels of Significance, p-Values and Standard Deviations are investigated. The solution is detailed and well presented.