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 14 Democrats being placed on the first line. The probability of getting a result as low as 14 is 0.029792.

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to determine whether the sample result of 14 Democrats being placed on the first line of voting ballots is statistically significant, given the probability of obtaining such a result is 0.029792.
2. Step 2: Recall the concept of statistical significance. A result is typically considered statistically significant if the probability (p-value) of obtaining it is less than a commonly used threshold, such as 0.05 (5%).
3. Step 3: Compare the given probability (0.029792) to the significance threshold (0.05). If the probability is less than 0.05, the result is statistically significant; otherwise, it is not.
4. Step 4: Interpret the comparison. If the probability is less than 0.05, it suggests that the observed result (14 Democrats on the first line) is unlikely to have occurred by random chance alone, indicating statistical significance.
5. Step 5: Conclude whether the sample result is significant based on the comparison. Ensure that the reasoning aligns with the threshold and the given probability value.

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 the 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 unlikely to have occurred by random chance.

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 calculate a test statistic and corresponding p-value. The results help determine whether to reject the null hypothesis in favor of the alternative, based on the significance level and the p-value.