Let's go through the steps to solve this problem:
### 1) Determine the null and alternative hypotheses.
We want to determine if the proportion of people who experienced dizziness varies between different treatment groups.
- Null Hypothesis [tex]\(H_0\)[/tex]: The proportion of people within each treatment group who experienced dizziness is the same.
- Alternative Hypothesis [tex]\(H_a\)[/tex]: The proportion of people within each treatment group who experienced dizziness is different.
Therefore, the correct pair of hypotheses is:
[tex]\[ H_0 : \text{The proportion of people within each treatment group who experienced dizziness are the same} \][/tex]
[tex]\[ H_a : \text{The proportion of people within each treatment group who experienced dizziness are different} \][/tex]
### 2) Determine the test statistic.
The test statistic used here is the Chi-Square statistic. The Chi-Square statistic is calculated as:
[tex]\[ \chi^2 = 4.67 \][/tex]
### 3) Determine the p-value.
The p-value associated with the test statistic is a measure of the probability that the observed differences occurred by chance alone under the null hypothesis. The p-value calculated is:
[tex]\[ \text{P-value} = 0.3226 \][/tex]
### 4) Make a decision.
To make a decision, we compare the p-value to the significance level, [tex]\(\alpha\)[/tex]. Typically, [tex]\(\alpha\)[/tex] is set at 0.05 (5%).
- If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis.
- If the p-value is greater than or equal to [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis.
In this case, the p-value is 0.3226, which is greater than 0.05. Hence, we fail to reject the null hypothesis.
### Decision:
Based on the p-value, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the proportion of people who experienced dizziness differs significantly among the treatment groups.
