3.2 Null hypothesis  (Page 3/3)

characteristics of 400 offspring. You note that there are 305 tall pea plants and 95 short pea plants; these are your  observed  values. Meanwhile, you  expect  that there will be 300 tall plants and 100 short plants from the Mendelian ratio.

You are now ready to perform statistical analysis of your results, but first, you have to choose a critical value at which to reject your null hypothesis. You opt for a critical value probability of 0.01 (1%) that the deviation between the observed and expected values is due to chance. This means that if the probability is less than 0.01, then the deviation is significant and not due to chance, and you will reject your null hypothesis. However, if the deviation is greater than 0.01, then the deviation is not significant and you will not reject the null hypothesis.

So, should you reject your null hypothesis or not? Here's a summary of your observed and expected data:

Now, let's calculate Pearson's chi-square:

For tall plants: Χ ²  = (305 - 300) ²  / 300 = 0.08

For short plants: Χ ²  = (95 - 100) ²  / 100 = 0.25

The sum of the two categories is 0.08 + 0.25 = 0.33

Therefore, the overall Pearson's chi-square for the experiment is Χ ²  = 0.33

Next, you determine the probability that is associated with your calculated chi-square value. To do this, you compare your calculated chi-square value with theoretical values in a chi-square table that has the same number of degrees of freedom. Degrees of freedom represent the number of ways in which the observed outcome categories are free to vary. For Pearson's chi-square test, the degrees of freedom are equal to  n  - 1, where  n  represents the number of different expected phenotypes (Pierce, 2005). In your experiment, there are two expected outcome phenotypes (tall and short), so  n  = 2 categories, and the degrees of freedom equal 2 - 1 = 1. Thus, with your calculated chi-square value (0.33) and the associated degrees of freedom (1), you can determine the probability by using a chi-square table (Table 1).

(Table adapted from Jones, 2008)

Note that the chi-square table is organized with degrees of freedom (df) in the left column and probabilities (P) at the top. The chi-square values associated with the probabilities are in the center of the table. To determine the probability, first locate the row for the degrees of freedom for your experiment, then determine where the calculated chi-square value would be placed among the theoretical values in the corresponding row.

At the beginning of your experiment, you decided that if the probability was less than 0.01, you would reject your null hypothesis because the deviation would be significant

and not due to chance. Now, looking at the row that corresponds to 1 degree of freedom, you see that your calculated chi-square value of 0.33 falls between 0.016, which is associated with a probability of 0.9, and 2.706, which is associated with a probability of 0.10. Therefore, there is between a 10% and 90% probability that the deviation you observed between your expected and the observed numbers of tall and short plants is due to chance. In other words, the probability associated with your chi-square value is much greater than the critical value of 0.01. This means that we will reject our null hypothesis, and the deviation between the observed and expected results is not significant.

Level of significance

Determining whether to accept or reject a hypothesis is decided by the experimenter, who is the person who chooses the "level of significance" or confidence. Scientists commonly use the 0.05, 0.01, or 0.001 probability levels as cut-off values. For instance, in the example experiment, you used the 0.01 probability. Thus, P ≥ 0.01, which can be interpreted to mean that chance likely caused the deviation between the observed and the expected values (i.e. there is a greater than 1% probability that chance explains the data). If instead we had observed that P ≤ 0.01, this would mean that there is less than a 1% probability that our data can be explained by chance. There is a significant difference between our expected and observed results, so the deviation must be caused by something other than chance.

References and recommended reading

Harris, J. A. A simple test of the goodness of fit of Mendelian ratios.  American Naturalist   46 , 741–745 (1912)

Jones, J. "Table: Chi-Square Probabilities." http://people.richland.edu/james/lecture/m170/tbl-chi.html (2008) (accessed July 7, 2008)

McDonald, J. H. Chi-square test for goodness-of-fit. From  The Handbook of Biological Statistics . http://udel.edu/~mcdonald/statchigof.html (2008) (accessed June 9, 2008)

Pearson, K. On the criterion that a given system of deviations from the probable in the case of correlated system of variables is such that it can be reasonably supposed to have arisen from random How Math Merged with Biology