Question

a sample of registered voters were asked if they supported a carbon tax to help government mitigate the harmful effects of global warming. the table below shows the results along party lines. answer questions 1 to 6 below by selecting the correct option. where relevant, all answers are listed to the nearest second decimal place. question 1 (1 point) if we took many, many samples of the same number of democrats as in the current sample, the sample proportions of their \yes\ votes would be distributed as a: not as a normal distribution as the standard assumptions are not met normal distribution with a mean of 37.56% and standard deviation of 3.32% normal distribution with a mean of 54.49% and standard deviation of 3.99% normal distribution with a mean of 76.23% and standard deviation of 2.62%

Explanation:

Step1: Calculate the proportion of "Yes" votes for Democrats

The total number of Democrats is \(63 + 202=265\). The proportion of "Yes" votes \(p=\frac{202}{265}\approx0.7623 = 76.23\%\)

Step2: Check the normal - distribution assumptions

For a sampling distribution of sample proportions \(\hat{p}\), the assumptions \(np\geq5\) and \(n(1 - p)\geq5\) must be met. Here \(n = 265\), \(np=265\times0.7623\approx202\geq5\) and \(n(1 - p)=265\times(1 - 0.7623)=265\times0.2377\approx63\geq5\)

Step3: Calculate the standard deviation of the sampling distribution of the sample proportion

The formula for the standard deviation of the sampling distribution of the sample proportion is \(\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}\)
Substitute \(p = 0.7623\) and \(n = 265\)
\(\sigma_{\hat{p}}=\sqrt{\frac{0.7623\times(1 - 0.7623)}{265}}=\sqrt{\frac{0.7623\times0.2377}{265}}\approx\sqrt{\frac{0.1813}{265}}\approx\sqrt{0.000684}\approx0.0262 = 2.62\%\)

Answer:

Normal distribution with a mean of \(76.23\%\) and standard deviation of \(2.62\%\)