Question

Exercises 7.9 to 7.12 give a null hypothesis for a goodness-of-fit test and a frequency table from a sample. For each table, find: (a) The expected count for the category labeled B. (b) The contribution to the sum of the chi-square statistic for the category labeled \(\mathrm{B}\). (c) The degrees of freedom for the chi-square distribution for that table. $$ \begin{aligned} &H_{0}: p_{a}=p_{b}=p_{c}=p_{d}=0.25\\\ &H_{a}:\\\ &\text { Some } p_{i} \neq 0.25\\\ &\begin{array}{lccc|c} \hline \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\\ \text { 40 } & 36 & 49 & 35 & 160 \\ \hline \end{array} \end{aligned} $$

Short answer

The expected count for category B is 40. The contribution to the chi-square statistic for category B is 0.4. The degrees of freedom for the chi-square distribution is 3.

Step-by-step solution

Calculate the Expected Count for Category B

The expected count for each of the four categories (A, B, C, D) under the null hypothesis is given by the total count times the probability for each category. As given by the null hypothesis, the probability for each category is 0.25. Therefore, the expected count for category B is \(0.25 \times 160 = 40\).

Calculate the Contribution to the Chi-Square Statistic for Category B

The contribution to the sum of the chi-square statistic from category B is given by \((Observed - Expected)^2 / Expected\). The observed count for category B is 36. Given that the expected count calculated in Step 1 was 40, the contribution from category B to the chi-square statistic is \((36 - 40)^2 / 40 = 0.4\).

Calculate the Degrees of Freedom

The degrees of freedom for the chi-square distribution is given by the number of categories minus one. In this case, there are four categories (A, B, C, D), meaning there are \(4 - 1 = 3\) degrees of freedom.

Key concepts

Expected Count

When conducting a goodness-of-fit test, the concept of expected count plays a pivotal role. It represents the frequency that is anticipated for each category if the null hypothesis holds true. In simple terms, the expected count is the count that would occur if the experimental data perfectly aligned with the probability distribution specified by the null hypothesis.

In our textbook exercise, the null hypothesis suggests that each of the categories A, B, C, and D are equally likely, each with a probability of 0.25. Given a total sample size of 160, the expected count for category B (as well as for the others) is calculated by multiplying the total count by the probability for that category, which is 0.25 in this case. Mathematically, it's expressed as the total count (160) times the probability (0.25), yielding an expected count of 40 for category B.

This count is essential because it provides a benchmark against which the observed count is compared. If the observed count deviates significantly from the expected count, it could indicate that the null hypothesis may not be an accurate representation of the underlying distribution. Thus, calculating the expected count is the first step in evaluating the goodness-of-fit.

Chi-Square Statistic

The chi-square statistic is a measure that quantifies how the observed counts deviate from the expected counts in a categorical data set. It is a crucial element of the goodness-of-fit test, helping statisticians to determine whether there is a statistically significant difference between the expected and observed frequencies in data categories.

To calculate the contribution of each category to the chi-square statistic, we use the formula \( (Observed - Expected)^2 / Expected \). For category B in our exercise, the observed count is 36, while the expected count is 40. Plugging these values into our formula, we obtain \( (36 - 40)^2 / 40 = 0.4 \), suggesting a discrepancy between what was observed in the sample and what we would expect under the null hypothesis.

The chi-square statistic is summed over all categories, and if the overall value is sufficiently large, this could lead to rejection of the null hypothesis, suggesting that the observed data does not fit the expected distribution. Interpretation of the chi-square value always takes into consideration the degrees of freedom and is referenced against a chi-square distribution table to determine statistical significance.

Degrees of Freedom

Degrees of freedom is a concept tied to the reliability of various statistical parameters and distributions, including the chi-square statistic. In the context of a goodness-of-fit test, the degrees of freedom define the number of values that can vary freely for a given statistic, after certain constraints (such as the total sum of observed counts) are taken into account.

For our chi-square test example, the degrees of freedom are determined by the number of categories minus one. Hence, with four categories (A, B, C, and D), the degrees of freedom for the test would be \(4 - 1 = 3\). It's important to note that the ‘minus one’ is to account for the restriction that all the expected frequencies must add up to the total sample size. We can essentially say that the degrees of freedom reflect the amount of ‘wiggle room’ you have in your data.

The degrees of freedom help to identify the correct distribution for evaluating the chi-square statistic and therefore, play a critical role in interpreting the test result. A chi-square distribution varies depending on the degrees of freedom, affecting how the critical value for statistical significance is determined. Therefore, correctly identifying the degrees of freedom is essential to conducting an accurate goodness-of-fit test.