Perception and Reality In a presidential election, 308 out of 611 voters surveyed said that they voted for the candidate who won (based on data from ICR Survey Research Group). Use a 0.05 significance level to test the claim that among all voters, the percentage who believe that they voted for the winning candidate is equal to 43%, which is the actual percentage of votes for the winning candidate. What does the result suggest about voter perceptions?

검증된 단계별 안내

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that the percentage of voters who believe they voted for the winning candidate is equal to 43% (p = 0.43). The alternative hypothesis (H₁) states that the percentage is not equal to 43% (p ≠ 0.43).

Step 2: Calculate the test statistic using the formula for a proportion z-test:

z = (p - p

₀)p₀(1-p₀)n, where p is the sample proportion, p₀ is the hypothesized proportion, and n is the sample size.

Step 3: Compute the sample proportion (p) using the formula

\frac{x}{n}

, where x is the number of voters who said they voted for the winning candidate (308) and n is the total number of voters surveyed (611).

Step 4: Determine the critical z-value for a two-tailed test at a significance level of 0.05. Look up the z-value corresponding to 0.025 in each tail (since 0.05 is split between two tails) in a standard normal distribution table.

Step 5: Compare the calculated z-test statistic to the critical z-value. If the test statistic falls outside the range defined by the critical z-values, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of voter perceptions.

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주요 개념

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Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population 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 this case, the null hypothesis would state that the percentage of voters who believe they voted for the winning candidate is 43%, while the alternative would suggest it is not.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a result is statistically significant. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In this scenario, if the p-value from the hypothesis test is less than 0.05, it would suggest that the observed voter perception significantly differs from the claimed percentage.

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 observing the sample data, or something more extreme, if the null hypothesis is true. A low p-value (typically less than the significance level) indicates strong evidence against the null hypothesis, suggesting that the actual voter perception may differ from the claimed 43%.