Does the candidate have more than 50% support?

Question

A campaign wants evidence that more than half of voters support its candidate. A poll of 600 voters finds 324 in favor (54%).

Run a one-sided one-proportion z-test at the 5% level (is support above 50%?): report the z-statistic and the decision.

Accepted Answer

The claim is directional (support exceeds 50%), so use an upper-tail test. Hypotheses: H0: p = 0.50 versus H1: p 0.50. Sample proportion: \hat p = 324/600 = 0.54. The standard error is computed under the null (p = 0.50): 	ext{SE} = \sqrt{\frac{p0(1-p0)}{n}} = \sqrt{\frac{0.50 	imes 0.50}{600}} = \sqrt{0.00041667} = 0.02041. Test statistic: z = \frac{\hat p - p0}{	ext{SE}} = \frac{0.54 - 0.50}{0.02041} = \frac{0.04}{0.02041} = 1.96. The one-sided 5% critical value is z{0.05} = 1.645. Since 1.96 1.645, reject H0. The one-sided p-value is 1 - \Phi(1.96) = 0.025. Conclusion: there is evidence …

Does the candidate have more than 50% support?

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