Question

A town politician is claiming they will receive more than 50% of the votes (and therefore win) in an upcoming election. An independent poll randomly sampled n=428 residents and 208 of them said they would vote for this politician. a. If you were to conduct a test to verify the claim made by the politician: i. Set up the null and alternative hypotheses. ii. Calculate the test statistic. Show your work. iii. Compared against a significance level of α=0.1, is the politician making a false claim? Why or why not? Explain your reasoning. b. What assumption(s) need to be made for the analysis in part (a) to be valid? Explain.

Answer 1

To test the politician's claim, we set up null and alternative hypotheses and calculated the test statistic as z ≈ -0.82. Considering the significance level of α = 0.1, we fail to reject the null hypothesis, indicating insufficient evidence to support the claim. The analysis assumes random sampling, a large enough sample size, and Bernoulli trials.

Step-by-step explanation:

To verify the politician's claim using hypothesis testing, we can set up the following hypotheses:

Step 1: Establish Hypotheses

• Null hypothesis (H0): The politician will receive 50% or fewer of the votes, i.e., p ≤ 0.50.
• Alternative hypothesis (Ha): The politician will receive more than 50% of the votes, i.e., p > 0.50.

Step 2: Calculate the Test Statistic

The test statistic for a proportion is found using the formula:

z = (p_hat - p0) / sqrt(p0 * (1 - p0) / n)

Here,

• p_hat = 208/428 (sample proportion)
• p0 = 0.50 (hypothesized proportion)
• n = 428 (sample size)

Calculating the test statistic gives us:

z = (208/428 - 0.50) / sqrt(0.50 * (1 - 0.50) / 428)

z ≈ -0.82

Step 3: Conclusion Using Significance Level α = 0.1

Since the z-score is negative, it falls to the left of the mean. A significance level of α = 0.1 corresponds to a z-score critical value of approximately 1.282 for a one-tailed test to the right. Because the calculated z-score does not exceed this critical value, we fail to reject the null hypothesis. Therefore, based on this sample, there is not enough evidence to support the politician's claim that they will receive more than 50% of the votes.

Step 4: Assumptions

The assumptions that need to be met for this analysis to be valid are:

• Random sampling of the residents.
• The sample size should be large enough for the central limit theorem to apply, which is generally satisfied in this case as n = 428.
• Bernoulli trials, implying residents can only vote 'for' or 'against' the politician.
• The sample proportion should be a good estimate of the true population proportion.