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Yesterday (old test question) there was a referendum in Italy. The measure would only count if 50% of the voters turned out to vote. The measure passed. But unfortunately for its backers, the turnout was only 49.6%. This percentage is not a sample, because all the ballots were counted. If instead of counting all the ballots, you had to determine the percentage of Yes votes with a sample, how large a sample would you have to take to get the result within 0.2% at 95% confidence? (0.3 points)

Respuesta :

Answer:

The required sample size is 240,085.

Step-by-step explanation:

The confidence interval for population proportion is:

[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex]

The margin of error in the interval is:

[tex]MOE=z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex]

Given:

MOE = 0.002

[tex]\hat p[/tex] = p = 0.496

[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]

*Use the z-table for the critical value.

Compute the sample size as follows:

[tex]MOE=z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n} }\\n=\frac{z_{\alpha/2}^{2}\times\hat p(1-\hat p)}{MOE^{2}} \\=\frac{(1.96)^{2}\times0.496\times(1-0.496)}{0.002^{2}} \\=240084.6336\\\approx240085[/tex]

Thus, the required sample size is 240,085.

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