Question

Suppose you wish to test the null hypothesis that three binomial parameters \(p_{A}, p_{B},\) and \(p_{c}\) are equal versus the alternative hypothesis that at least two of the parameters differ. Independent random samples of 100 observations were selected from each of the populations. Use the information in the table to answer the questions in Exercises \(5-7 .\) $$ \begin{array}{lrrrr} \hline & {\text { Population }} & \\ & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \text { Successes } & 24 & 19 & 33 & 76 \\ \text { Failures } & 76 & 81 & 67 & 224 \\ \hline \text { Total } & 100 & 100 & 100 & 300 \end{array} $$ Use the approximate \(p\) -value to determine the statistical significance of your results. If the results are statistically significant, explore the nature of the differences in the three binomial proportions.

Short answer

Answer: The pooled proportion is calculated by dividing the total number of successes by the total number of observations. In this specific case, the pooled proportion would be calculated as 76/300.

Step-by-step solution

Calculate the Pooled Proportion

The pooled proportion (\(\tilde{p}\)) is obtained by dividing the total number of successes by the total number of observations. In this case: $$\tilde{p} = \frac{\text{Total Successes}}{\text{Total Observations}} = \frac{76}{300}$$

Compute the Expected Values for A, B, and C

Calculate the expected values for each population using their respective total observations and the pooled proportion: $$\text{Expected Value}_A = n_A \times \tilde{p} = 100 \times \frac{76}{300}$$ $$\text{Expected Value}_B = n_B \times \tilde{p} = 100 \times \frac{76}{300}$$ $$\text{Expected Value}_C = n_C \times \tilde{p} = 100 \times \frac{76}{300}$$

Calculate the Chi-Squared Statistic

To find the chi-squared statistic, use the formula: $$\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$$ Calculate the chi-squared statistic for A, B, and C: $$\chi^2_A = \frac{(24 - 100 \times \frac{76}{300})^2}{100 \times \frac{76}{300}}$$ $$\chi^2_B = \frac{(19 - 100 \times \frac{76}{300})^2}{100 \times \frac{76}{300}}$$ $$\chi^2_C = \frac{(33 - 100 \times \frac{76}{300})^2}{100 \times \frac{76}{300}}$$ Then sum up the chi-squared values to obtain the overall chi-squared statistic: $$\chi^2 = \chi^2_A + \chi^2_B + \chi^2_C$$

Calculate the Approximate P-Value

Using the chi-squared statistic, we can find the approximate p-value. For this, we need a chi-squared table with a significance level (e.g., 0.05) and degrees of freedom, which is \((3-1) = 2\). Look up the corresponding critical value in the table and compare it to our calculated chi-squared statistic. If our chi-squared statistic is greater than the critical value, then the null hypothesis should be rejected.

Explore the Differences (If Statistically Significant)

If the results are statistically significant, meaning that the null hypothesis is rejected, we can explore the nature of the differences between the proportions. Compare the observed proportions for each population, A, B, and C, to determine where the differences lie. Consider pairwise comparisons using a post-hoc test, such as Tukey's HSD test, to find out which proportions are significantly different from one another.

Key concepts

Pooled Proportion

The pooled proportion is a combined estimate of the expected success rate across several groups. It is particularly useful in hypothesis testing when we want to determine if different groups have the same probability of success. To calculate the pooled proportion, we sum all the successes from every group and divide by the total number of observations across these groups.

For example, in a chi-squared test for binomial parameters, as seen in the given exercise, we have groups A, B, and C with a total of 76 successes out of 300 observations, resulting in a pooled proportion of \( \tilde{p} = \frac{76}{300} \) for all groups. This value assumes the null hypothesis is true and the success rates are the same across all groups, serving as a baseline for the expected frequencies.

Expected Value Calculation

Expected value plays a central role in the chi-squared test. It represents what we would anticipate under the null hypothesis. To compute expected values in our context, we multiply the total observations in each group by the pooled proportion. This gives us a benchmark to compare against the observed successes.

In our example, we would calculate the expected successes in each group A, B, and C using \( \text{Expected Value}_A = n_A \times \tilde{p} \) and so on. These values are crucial because they are used in the calculation of the chi-squared statistic, which is the main driver in determining whether to accept or reject our initial hypothesis.

Chi-Squared Statistic

The chi-squared statistic quantifies the discrepancy between observed and expected frequencies. It is the core of the chi-squared test and helps us decide if the differences we see could be due to random chance or if they are statistically significant.

In calculating the chi-squared statistic, each observed frequency is compared to its expected counterpart, and the sum of these comparisons across all categories is our chi-squared value. Mathematically, it's denoted as \( \chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}} \). A higher chi-squared value indicates a larger discrepancy between observed and expected results and suggests that the null hypothesis may not hold.

P-Value Significance

The p-value signifies the probability of obtaining a test statistic as extreme as the one computed, assuming the null hypothesis is true. A small p-value indicates that such an extreme observed result is unlikely due to random chance alone, leading to questioning the validity of the null hypothesis.

Typically, a significance level (\( \alpha \) value) is chosen, such as 0.05. If our p-value is less than this \( \alpha \) value, we reject the null hypothesis, indicating a statistically significant difference in the binomial parameters. The calculation of the p-value involves comparing our chi-squared statistic to a chi-squared distribution with the appropriate degrees of freedom related to our test.

Null Hypothesis in Statistics

The null hypothesis is the default assumption that there is no effect or no difference in a statistical hypothesis test. It serves as the assertion we aim to challenge with our data. In binomial parameter tests, the null hypothesis typically posits that the probabilities of success in different groups are equal.

In the provided exercise, the null hypothesis is that the binomial parameters \( p_{A}, p_{B}, \) and \( p_{C} \) are the same for all groups. Through the chi-squared test, we assess if the observed data provide enough evidence to reject this assumption. Rejecting the null hypothesis suggests that at least two of the population proportions differ, which would prompt further investigation into the nature of these differences.