Question

The appearance of leaf pigment glands in the seedling stage of cotton plants is genetically controlled. According to one theory of the control mechanism, the population ratio of glandular to glandless plants resulting from a certain cross should be \(11: 5 ;\) according to another theory it should be \(13: 3 .\) In one experiment, the cross produced 89 glandular and 36 glandless plants. \({ }^{45}\) Use goodness-of-fit tests (at \(\alpha=0.10\) ) to determine whether these data are consistent with (a) the 11: 5 theory (b) the 13: 3 theory

Short answer

To determine which theory is consistent with the data, calculate the chi-square test statistic for both the 11:5 and 13:3 ratios and compare them to the critical value at a significance level of 0.10. If the test statistic is less than the critical value, the corresponding theory is considered consistent with the observed data.

Step-by-step solution

State Null and Alternative Hypotheses

For both theories, we need to formulate the null hypothesis (H0) and the alternative hypothesis (H1). \(H0\): The observed data fits the expected ratio. \(H1\): The observed data does not fit the expected ratio.

Calculate Expected Frequencies for the 11:5 Theory

To calculate expected frequencies under the 11:5 theory, we first need to find the total number of plants, and then apply the 11:5 ratio. The total is 89 glandular + 36 glandless = 125 plants. Expected glandular = \(\frac{11}{16} \times 125\), Expected glandless = \(\frac{5}{16} \times 125\).

Calculate Chi-square Test Statistic for the 11:5 Theory

Use the formula \(\chi^2 = \sum \frac{(O - E)^2}{E}\), where O = observed frequency and E = expected frequency. Calculate this for each category, glandular and glandless, and sum them to find the test statistic.

Calculate Expected Frequencies for the 13:3 Theory

Following the same method in Step 2, calculate the expected frequencies for the 13:3 theory. Expected glandular = \(\frac{13}{16} \times 125\), Expected glandless = \(\frac{3}{16} \times 125\).

Calculate Chi-square Test Statistic for the 13:3 Theory

Similar to Step 3, use the same \(\chi^2\) formula to calculate the test statistic for the 13:3 theory using the observed and new expected frequencies.

Determine Degrees of Freedom

For both theories, the degrees of freedom (df) = number of categories - 1. Since there are two categories (glandular and glandless), df = 2 - 1 = 1.

Find Chi-square Critical Values

Using a chi-square table and the significance level \(\alpha = 0.10\), find the critical value of the chi-square distribution corresponding to df = 1.

Make Decisions for the 11:5 and 13:3 Theories

If the test statistic calculated in Steps 3 and 5 is greater than the critical value found in Step 7, the null hypothesis is rejected. Otherwise, the null hypothesis cannot be rejected.

Key concepts

Chi-Square Test Statistic

Understanding the chi-square test statistic is essential when conducting a goodness-of-fit test. This statistic helps us determine how well our observed data match what we would expect theoretically. In essence, the chi-square test is designed to analyze the discrepancies between observed and expected frequencies under a given hypothesis.

The calculation involves the formula \(\chi^2 = \sum \frac{(O - E)^2}{E}\), where \(O\) represents the observed frequency, and \(E\) is the expected frequency for each category. We square the difference between the observed and expected values to ensure that all discrepancies are positive and weighted by how significant they are relative to the expected frequency. The summation symbol \(\sum\) indicates we must add up the values calculated for each category. This sum yields the chi-square test statistic.

Interpreting the Chi-Square Statistic

Once calculated, we compare the test statistic to a critical value from the chi-square distribution table, which is determined by the degrees of freedom and our chosen significance level. If the test statistic is higher than the critical value, we have grounds to reject the null hypothesis, indicating our observed data do not fit the expected pattern well. On the other hand, if the test statistic is lower, we cannot confidently say there is a significant difference; hence, we fail to reject the null hypothesis.

Null Hypothesis in Statistics

The null hypothesis is the default assumption that there is no relationship between two measured phenomena. In the context of a chi-square goodness-of-fit test, it posits that there is no significant difference between the observed frequencies and the expected frequencies derived from a theoretical distribution.

To test a null hypothesis, we contrast it with an alternative hypothesis which suggests that there is indeed a difference or relationship. As we perform a chi-square test, we start with the presumption that the null hypothesis is true. Only after computing our test statistic and comparing it to the critical value, can we consider if there's enough evidence to challenge this initial position.

Role in Decision-Making

The null hypothesis serves as a benchmark for statistical significance. If our evidence (the calculated chi-square test statistic) is strong enough to surpass a threshold (the critical chi-square value), the null hypothesis is rejected. It's crucial to note that not rejecting the null hypothesis does not prove its truth. Instead, it simply indicates that our sample data do not provide sufficient evidence to dispute it.

Expected Frequency Calculation

Expected frequency calculation is vital in chi-square tests as it establishes the theoretical frequencies of outcomes based on a proposed hypothesis. These expected frequencies are then used to compare against the actual observed frequencies.

To calculate expected frequencies, you must first understand the theoretical probability distribution at play. For instance, in genetics, expected frequencies for traits are often calculated based on Mendelian ratios or other theoretical genetic distributions. Using the total sample size and the expected probability of occurrence for each outcome, you can determine the expected frequency for each category: \(E = Probability \times Total \ Sample \ Size\).

Practical Application

In our example, we calculated the expected numbers of glandular and glandless plants by multiplying the total number of plants by the probabilities from the two proposed theories. This step forms the foundation of our goodness-of-fit test, as these expected values are used directly in the computation of the chi-square test statistic. A precise calculation of expected frequencies is paramount as inaccuracies here could lead to incorrect conclusions in the hypothesis-testing process.