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In any statistical test, the null hypothesis is a crucial starting point. It is a statement that what we expect to observe is actually true. In the context of our chi-squared test involving corn seedlings, the null hypothesis, denoted as \(H_{0}\), states that the observed distribution of green and yellow seedlings is consistent with theoretical expectations.
Specifically, the null hypothesis suggests that 75% of the seedlings are expected to be green. This implies \(H_{0}: p = 0.75\) for green seedlings. The purpose of setting up this hypothesis is to test whether the observed data (854 green seedlings) align with these expectations. Rejecting \(H_{0}\) would mean there is evidence that the observed and expected frequencies differ significantly.
Thus, we inspect the alignment between what we see and what we expect as per our model. If they align well, our null hypothesis stands. If not, it may be time to rethink our underlying assumptions or model.

Expected frequencies are the predicted counts of each category based on a predefined model, such as a theoretical probability distribution. In our experiment with corn seedlings, expected frequencies are computed by multiplying the total number of seedlings by the expected proportion of each category.
For green seedlings, the expected frequency \(E_{\text{Green}}\) is found by multiplying 75% by the total 1,103 seedlings, which equals 827.25. Meanwhile, for yellow seedlings, the percentage is 25%, giving an expected count of 275.75. These calculations allow us to compare these theoretical values to the actual observed frequencies of 854 and 249, respectively.
The primary purpose of calculating expected frequencies is to have a baseline to analyze discrepancies between observed and expected data. If discrepancies are significant, it suggests that the assumption made by the theoretical model (like the null hypothesis) may not hold.

Degrees of freedom ( extit{df}) is a concept used to determine the number of values in a statistical calculation that are free to vary. It is an integral part of chi-squared tests and impacts the critical value needed to reject the null hypothesis.
In a chi-squared test, the formula for degrees of freedom is \(df = k - 1\), where \(k\) is the total number of categories. For our corn seedling example, we have two categories: green and yellow. Thus, \(df = 2 - 1 = 1\). Understanding degrees of freedom helps us assess how much information is available to estimate another parameter or fit a model.
The degrees of freedom calculated plays a role in determining the P-value. Different degrees of freedom correspond to different probability distributions, underscoring why this concept is fundamental in making statistical inferences.

The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the statistic obtained, given that the null hypothesis is true. It helps in deciding whether to reject the null hypothesis.
For our corn seedling data, we calculated a chi-squared statistic of 13.474 with one degree of freedom. From chi-squared distribution tables or computational tools, we obtain a P-value of less than 0.001. This very low P-value suggests that the observed data differs significantly from what the null hypothesis predicts.
When the P-value is less than a chosen significance level, often 0.05, we reject the null hypothesis. In this situation, the P-value tells us that the likelihood of observing such a data set if our theoretical model is true is very small. Hence, we conclude there is strong evidence that our observed distribution of seedling colors does not fit the expected model.