Question

A geneticist self-pollinated pink-flowered snapdragon plants and produced 97 progeny with the following colors: 22 red plants, 52 pink plants, and 23 white plants. \(^{16}\) The purpose of this experiment was to investigate a genetic model that states that the probabilities of red, pink, and white are \(0.25,0.50,\) and \(0.25 .\) Identify the type of statistical method that is appropriate for these data, but do not actually conduct the analysis.

Short answer

The Chi-Square Goodness of Fit test is the appropriate statistical method to assess the fit between observed and expected probabilities of categorical outcomes in this genetic experiment.

Step-by-step solution

Identify the Problem

The problem requires checking if the observed frequencies of the colors of snapdragon plants fit a certain genetic model. The proposed model suggests that the probability distribution for the colors is as follows: red (0.25), pink (0.50), and white (0.25).

Choose the Appropriate Statistical Test

To compare the observed frequencies with the expected frequencies, the Chi-Square Goodness of Fit test is the appropriate statistical method. This test is used to determine how well the observed data fits a theoretical model.

Justification for the Selected Test

The Chi-Square Goodness of Fit test is selected because it is designed to test the hypothesis that the observed frequency distribution of a nominal variable matches an expected distribution. Since the problem involves categorical data (plant colors) and a comparison with expected probability distributions, this test is suitable.

Key concepts

Statistical Hypothesis Testing

Statistical hypothesis testing is a core component of many scientific studies, including genetics. It involves making an assumption, called the null hypothesis, and then determining if the data collected provides sufficient evidence to reject this hypothesis. The alternative hypothesis is what researchers typically hope to demonstrate.

In the context of our exercise involving snapdragon plants, the null hypothesis asserts that the observed color distribution among the progeny matches the theoretical genetic model. Conversely, the alternative hypothesis would suggest that the observed distribution significantly deviates from expectations. The purpose of conducting a hypothesis test is to make a statistically-informed decision regarding the likelihood of this deviation being due to chance alone.

The outcome of a hypothesis test is often conveyed through a p-value, which quantifies the probability of observing a test statistic at least as extreme as the one calculated, assuming the null hypothesis is true. If this p-value is less than a chosen significance level, typically 0.05, statisticians consider it strong evidence against the null hypothesis.

Categorical Data Analysis

Categorical data analysis deals with datasets where the variables represent distinct categories or groups rather than numerical values. Data like the colors of plants are perfect examples, with categories being red, pink, and white.

When analyzing categorical data, one must use statistical methods appropriate for discrete outcomes, as standard methods for continuous data (like mean and variance) do not apply. Tools for categorical data analysis include frequency counts, contingency tables, and Chi-Square tests, among others.

Since the exercise provided involves categorical outcomes, analyzing the data requires a test designed for such cases—the Chi-Square Goodness of Fit test. This test allows us to compare the observed frequencies of plant colors with those expected under the genetic model. If the data significantly deviates from the model, something other than random genetic assortment may be happening.

Probability Distributions

Probability distributions are mathematical functions that describe how the probabilities of various outcomes are distributed in a system. These distributions allow us to express the expected frequency of events occurring in different categories.

For example, simple genetic inheritance patterns can often be described using discrete probability distributions because offspring traits, such as snapdragon flower color, fall into distinct categories. The genetic model outlined in the exercise specifies that the probabilities of the plant colors are fixed at 0.25 for red, 0.5 for pink, and 0.25 for white. This implies a theoretical distribution against which the observed outcomes can be compared.

Understanding probability distributions is crucial when using statistical tests like the Chi-Square Goodness of Fit. This test compares the expected frequencies, as determined by the genetic model's probability distribution, with actual observations from an experiment.

Genetic Models

Genetic models are theories or formulas that predict how traits will be inherited from one generation to the next. They are based on understanding how genes interact and segregate during reproduction.

In the context of the exercise, the genetic model predicts the probability of offspring exhibiting each possible flower color based on Mendelian genetics. The model suggests a 1:2:1 ratio for the colors red:pink:white, respectively. This ratio is a hallmark of simple Mendelian inheritance for a trait controlled by two alleles exhibiting incomplete dominance.

Using such genetic models in experimentation allows for testing the validity of proposed inheritance patterns by comparing observed data with those predicted by the model. Discrepancies between observed and expected outcomes might indicate additional genetic complexities or may warrant a reevaluation of the model itself.