Question

Geneticists studying the inheritance pattern of cowpea plants classified the plants in one experiment according to the nature of their leaves. The data follow : $$ \begin{array}{|lccc|} \hline \text { Type } & \text { I } & \text { II } & \text { III } \\ \hline \text { Number } & 179 & 44 & 23 \\ \hline \end{array} $$ Test the null hypothesis that the three types occur with probabilities \(12 / 16,3 / 16,\) and \(1 / 16 .\) Use a chi-square test with \(\alpha=0.10\).

Short answer

Calculate expected frequencies for each type, compute the chi-square statistic, compare it to the critical value at \(\alpha = 0.10\) and make a conclusion about the null hypothesis.

Step-by-step solution

- Determine Expected Frequencies

We are given the probabilities for each type as follows: Type I: \(\frac{12}{16}\), Type II: \(\frac{3}{16}\), and Type III: \(\frac{1}{16}\). To find the expected frequency for each type, multiply the total number of observations by the probability of each type. The sum of the observed numbers is 179 (Type I) + 44 (Type II) + 23 (Type III), which equals 246. The expected frequencies are \(\frac{12}{16} * 246 \) for Type I, \(\frac{3}{16} * 246 \) for Type II, and \(\frac{1}{16} * 246 \) for Type III.

- Calculate the Chi-square Statistic

The chi-square statistic is calculated using the formula \(\chi^2 = \Sigma \frac{(O - E)^2}{E}\), where \(O\) is the observed frequency, and \(E\) is the expected frequency for each class. Calculate the chi-square statistic separately for the three types and sum them up to get the total chi-square value.

- Determine the Degrees of Freedom

The degrees of freedom for this test are equal to the number of classes minus one. Therefore, with three types, the degrees of freedom are \(3 - 1 = 2\).

- Find the Critical Chi-square Value

Using a chi-square distribution table, find the critical value corresponding to the given alpha level of 0.10 and the determined degrees of freedom (2). This will give us the threshold we need to determine whether to reject the null hypothesis.

- Compare the Statistic and Critical Values

If the calculated chi-square statistic is greater than the critical value from the table, we reject the null hypothesis. If it is smaller, we do not reject the null hypothesis.

- Make a Conclusion

Based on the comparison between the chi-square statistic and the critical value, make a conclusion about the null hypothesis. State whether there is sufficient evidence to reject the hypothesis that the three types occur with the given probabilities.

Key concepts

Null Hypothesis

When performing any statistical test, we always start with a statement that there is no significant difference between observed data and expected data. This initial statement is known as the null hypothesis. In the context of the chi-square test from the given exercise, the null hypothesis is the presupposition that cowpea plants exhibit three types of leaves with the specific probabilities of 12/16, 3/16, and 1/16 respectively.

In simple terms, the null hypothesis sets the stage for testing by stating that any difference between your experiment's observations and what you expect to happen is just due to random chance. To challenge this assumption, we use statistics – if the results are highly unusual under the null hypothesis, we have grounds to reject it.

Expected Frequencies

The term expected frequencies comes into play when we want to compare what we observe in our data to what we would expect to see if the null hypothesis were true. These expected frequencies are not based on actual observations but on the proposed probabilities in the null hypothesis multiplied by the total number of observations.

For example, if we think that Type I cowpea plants should make up 75% of our sample, and we have 246 plants in total, we'd expect to see \( 0.75 \times 246 \text{ (which is \frac{12}{16} \times 246)} \) of them as Type I. The same calculation applies to Type II and Type III, using their respective probabilities. It's essential to correctly calculate these numbers because they're crucial for figuring out the chi-square statistic.

Degrees of Freedom

The term degrees of freedom is a bit less intuitive than expected frequencies or the null hypothesis, but it's just as crucial in any statistical test. It refers to the number of independent values that a statistical analysis can have without breaking any constraints. In the case of a chi-square test, the degrees of freedom are typically the number of categories minus one.

In our cowpea plant study, we have three types of leaves (three categories). Thus, the degrees of freedom is calculated as the number of types minus one, which is 3 - 1 = 2. The degrees of freedom help to determine which chi-square distribution to use when we want to find out how extreme our results are. This, in turn, guides us in making decisions about our null hypothesis based on the calculated chi-square value.