Question

In an agronomy experiment, peanuts with shriveled seeds were crossed with normal peanuts. The genetic model that the agronomists were considering predicted that the ratio of normal to shriveled progeny would be 3: 1 . They obtained 95 normal and 54 shriveled progeny. Do these data support the hypothesized model? Here is computer output for a chi-square test using a nondirectional alternative. \(X\) -squared \(=10.0425, d f=1,\) p-value \(=0.00153\) (a) State the null and alternative hypotheses in context. (b) Compute the expected frequencies. (c) If \(\alpha=0.01\), what is your conclusion regarding \(H_{0}\) ?

Short answer

The null hypothesis is that the ratio of normal to shriveled progeny is 3:1. The expected frequencies are 111.75 normal and 37.25 shriveled progeny. Since the p-value (0.00153) is less than the significance level (0.01), we reject the null hypothesis, indicating the data do not support the 3:1 ratio.

Step-by-step solution

State the Null and Alternative Hypotheses

In the context of the experiment, the null hypothesis (\(H_{0}\)) is that the ratio of normal to shriveled progeny is 3:1. The alternative hypothesis (\(H_{A}\)) suggests that the ratio is different from 3:1.

Compute Expected Frequencies

The total number of progeny is 95 normal + 54 shriveled = 149. Under the null hypothesis, a 3:1 ratio means we expect three normal progeny for every one shriveled progeny. Therefore, we expect \(\frac{3}{4}\times 149 = 111.75\) normal progeny and \(\frac{1}{4}\times 149 = 37.25\) shriveled progeny.

Use the chi-square test result

Given the chi-square test statistic is 10.0425 with a p-value of 0.00153 and a significant level (\(\alpha\)) of 0.01, we reject the null hypothesis because the p-value is less than \(\alpha\). The data do not support the 3:1 ratio hypothesized by the genetic model.

Key concepts

Genetic Model

In genetics, a genetic model is a hypothesis regarding the inheritance pattern of a particular trait based on the underlying genetics. These models are foundational in predicting how traits will be passed on to subsequent generations. For example, the simplest model for a single gene trait follows Mendelian inheritance patterns, predicting a 3:1 ratio for dominant to recessive traits in a monohybrid cross.

When agronomists, such as those involved in the peanut experiment, talk about a 3:1 genetic model, they are referencing this Mendelian ratio. It suggests that, assuming one trait is dominant over the other, about three-quarters of the progeny will exhibit the dominant phenotype (normal peanuts), and one-quarter will exhibit the recessive phenotype (shriveled seeds).

Null Hypothesis

The null hypothesis, denoted as H₀, is a statement used in statistical analyses that there is no effect or no difference, and it serves as the starting assumption for many tests, including the chi-square test. It's the hypothesis that researchers aim to disprove or fail to reject, based on the data collected.

In the context of the agronomy experiment, the null hypothesis specifically states that the genetic model predicting a 3:1 ratio of normal to shriveled peanuts is correct. The data obtained from actual experiments can be compared to this hypothesis to check its validity.

Alternative Hypothesis

The alternative hypothesis, denoted as H_A or H₁, is the complement to the null hypothesis in statistical testing. It represents the researcher's actual hypothesis, asserting that there is an effect or a difference. In contrast to the null hypothesis, the alternative hypothesis is what researchers often want to demonstrate as true.

For the peanut progeny experiment, the alternative hypothesis claims that the ratio of normal to shriveled progeny is different from the 3:1 ratio predicted by the genetic model. If the evidence is strong enough to reject the null hypothesis, it supports the alternative hypothesis.

Expected Frequencies

In the context of chi-square tests, expected frequencies are the predicted outcomes that would occur if the null hypothesis were true. These expected counts are crucial for comparing what the theory predicts with what the actual data shows.

To compute these expected frequencies, one must first establish the predicted ratio (such as 3:1 from the genetic model), then apply this ratio to the total observed data to partition it accordingly. In the peanut experiment, by multiplying the total number of observed progeny by the expected ratios (3/4 for normal and 1/4 for shriveled), we obtain the expected frequencies necessary for performing the chi-square test.

Statistical Significance

The concept of statistical significance is a measure of whether the observed effect or difference in data is likely due to chance. Analysts use a pre-set significance level, denoted as alpha (α), to help decide this. The p-value obtained from statistical tests is compared against this alpha level.

Commonly, alpha levels of 0.01 or 0.05 are used. If the p-value is lower than the alpha level, the result is statistically significant, and the null hypothesis is typically rejected, indicating support for the alternative hypothesis. In the peanut experiment, a p-value of 0.00153 is less than the chosen α level of 0.01, leading to the conclusion that the observed ratio significantly deviates from the predicted 3:1 ratio, thus providing evidence to reject the null hypothesis.