In Exercises 33–40, use the given probability value to determine whether the sample results are significant.




Voting Repeat Exercise 33 after replacing 40 Democrats being placed on the first line of voting ballots with 27 Democrats being placed on the first line. The probability of getting a result as high as 27 is 0.029792.

Verified step by step guidance

1. Identify the given probability value, which is 0.029792, and compare it to the commonly used significance level (α), such as 0.05 or 0.01. This will help determine whether the result is statistically significant.
2. Recall the rule for statistical significance: If the probability (p-value) is less than or equal to the significance level (α), the result is considered statistically significant. Otherwise, it is not significant.
3. State the null hypothesis (H₀) and the alternative hypothesis (H₁). For this problem, H₀ might state that the placement of Democrats on the first line is random, while H₁ might suggest that it is not random.
4. Compare the given probability (0.029792) to the chosen significance level (e.g., 0.05). If 0.029792 ≤ 0.05, reject the null hypothesis (H₀). If 0.029792 > 0.05, fail to reject H₀.
5. Conclude whether the result of 27 Democrats being placed on the first line is statistically significant based on the comparison. If significant, it suggests that the placement may not be random; if not significant, it suggests the placement could be due to chance.

Key Concepts

Significance Level

The significance level, often denoted as alpha (α), is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. Commonly set at 0.05, it represents the probability of making a Type I error, which occurs when a true null hypothesis is incorrectly rejected. In this context, if the probability value (p-value) is less than α, the results are considered statistically significant.

P-value

The p-value is a statistical measure that helps determine the strength of the evidence against the null hypothesis. It quantifies the probability of observing results as extreme as, or more extreme than, the actual observed results, assuming the null hypothesis is true. A low p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting that the observed effect is statistically significant.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0 in favor of H1. The process includes calculating the p-value and comparing it to the significance level to assess the validity of the hypotheses.