To test the hypothesis that at least 60% of voters favor the free trade agreement (FTA) based on the recent poll, we can perform a hypothesis test for proportions.
Step 1: State the null and alternative hypotheses.
- Null Hypothesis (H0): \( p \geq 0.60 \) (At least 60% of voters favor the FTA)
- Alternative Hypothesis (H1): \( p < 0.60 \) (Less than 60% of voters favor the FTA)
Step 2: Determine the sample proportion.
From the poll:
- Sample size \( n = 400 \)
- Number of individuals favoring the FTA \( x = 250 \)
The sample proportion \( \hat{p} \) is calculated as: \[ \hat{p} = \frac{x}{n} = \frac{250}{400} = 0.625 \]
Step 3: Calculate the test statistic.
We will use the formula for the test statistic for proportions: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Where:
- \( p_0 = 0.60 \) is the population proportion under the null hypothesis,
- \( n = 400 \) is the sample size.
First, we calculate the standard error: \[ \text{Standard Error} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.60 \times (1 - 0.60)}{400}} = \sqrt{\frac{0.60 \times 0.40}{400}} = \sqrt{\frac{0.24}{400}} = \sqrt{0.0006} \approx 0.02449 \]
Now, substitute \( \hat{p} \), \( p_0 \), and the standard error into the z formula: \[ z = \frac{0.625 - 0.60}{0.02449} \approx \frac{0.025}{0.02449} \approx 1.0205 \]
Step 4: Determine the critical value and make a decision.
For a one-tailed test at a significance level of \( \alpha = 0.05 \), we look up the critical z-value:
- The critical z-value for \( \alpha = 0.05 \) (one-tailed) is approximately -1.645 (since we are testing if the proportion is less than 0.60).
Now we compare our calculated z-value with the critical z-value:
- Calculated z-value: \( z \approx 1.0205 \)
- Critical z-value: -1.645
Step 5: Conclusion
Since \( z = 1.0205 \) is greater than -1.645, we do not reject the null hypothesis.
Final Decision
At the 5% significance level, there is insufficient evidence to conclude that less than 60% of voters favor the free trade agreement. Therefore, we accept the belief that at least 60% of voters favor the FTA based on the sample data.
1 answer