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




Voting Repeat the preceding Exercise 33 after replacing 40 Democrats being placed on the first line of voting ballots with 26 Democrats being placed on the first line. The probability of getting a result as high as 26 is 0.058638.

Verified step by step guidance

1. Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). In this case, the null hypothesis assumes that the placement of Democrats on the first line of the voting ballot is random, while the alternative hypothesis suggests that the placement is not random.
2. Determine the significance level (α). If not explicitly stated, a common choice is α = 0.05. This is the threshold for deciding whether the sample result is statistically significant.
3. Compare the given probability (p-value) of 0.058638 to the significance level (α). The p-value represents the probability of obtaining a result as extreme as 26 Democrats being placed on the first line, assuming the null hypothesis is true.
4. If the p-value is less than or equal to the significance level (p ≤ α), reject the null hypothesis and conclude that the result is statistically significant. If the p-value is greater than the significance level (p > α), fail to reject the null hypothesis and conclude that the result is not statistically significant.
5. State the conclusion in the context of the problem. For example, if the result is not statistically significant, you would conclude that there is insufficient evidence to suggest that the placement of Democrats on the first line is not random.

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 indicates a 5% risk of concluding that a difference exists when there is none. If the p-value (probability 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 represents the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. A lower p-value indicates stronger 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 determine whether to reject H0 in favor of H1. The outcome is influenced by the p-value and the chosen significance level, guiding researchers in drawing conclusions about their data.