When a simple random poll of 1000 people says that a presidential candidate has of the support in
particular state, does this mean that the candidate should be confident in her chances of winning the
electoral votes in that state? Why or why not?

Question

When a simple random poll of 1000 people says that a presidential candidate has

of the support in
particular state, does this mean that the candidate should be confident in her chances of winning the
electoral votes in that state? Why or why not?

UpStudy AI · Accepted Answer

**Step 1: Understand the Poll Result**

The poll sampled $$ n = 1000 $$ people and found that $$ 51\% $$ support the candidate. However, in any sample, there is random sampling error. This means the true level of support in the state might differ from the sample proportion because of the variabilities inherent in taking a sample rather than a census.

**Step 2: Compute the Standard Error**

For a proportion, the standard error (SE) is given by

$$
\text{SE} = \sqrt{\frac{p(1-p)}{n}}
$$

Here, $$ p = 0.51 $$ and $$ n = 1000 $$. Therefore,

$$
\text{SE} = \sqrt{\frac{0.51(1-0.51)}{1000}} = \sqrt{\frac{0.51 \times 0.49}{1000}} \approx \sqrt{\frac{0.2499}{1000}} \approx \sqrt{0.0002499} \approx 0.0158
$$

**Step 3: Construct a Confidence Interval**

Assuming a normal approximation, a 95% confidence interval is given by

$$
p \pm z \times \text{SE}
$$

where $$ z \approx 1.96 $$ for a 95% confidence level. This gives

$$
0.51 \pm 1.96 \times 0.0158 \approx 0.51 \pm 0.031
$$

So, the 95% confidence interval is approximately

$$
(0.479, 0.541)
$$

**Step 4: Interpret the Confidence Interval**

The interval $$(0.479, 0.541)$$ means that if we were to repeat the poll many times, we would expect the true proportion of support in the state to be within this range about 95% of the time. Notice that the lower bound is below $$50\%$$. Therefore, it is statistically plausible (according to this interval) that the candidate's true level of support might be less than a majority.

**Step 5: Conclusion on Electoral Confidence**

Given that the candidate's apparent lead of $$51\%$$ in the poll has a margin of error that could potentially bring her true support under $$50\%$$ (as seen with the lower bound of the confidence interval), the candidate should not be overly confident in winning the electoral votes solely based on this poll result. The possibility exists that the true support could be insufficient to secure victory in the state, particularly if the contest is close.

Thus, while the poll offers an indication, it does not guarantee a win due to the inherent uncertainty in sample estimates.
The poll shows the candidate has 51% support, but with a margin of error, the true support could be as low as 47.9%. Therefore, the candidate shouldn't be too confident about winning the state's electoral votes based solely on this poll.

When a simple random poll of 1000 people says that a presidential candidate has of the support in
particular state, does this mean that the candidate should be confident in her chances of winning the
electoral votes in that state? Why or why not?

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When a simple random poll of 1000 people says that a presidential candidate has of the support in particular state, does this mean that the candidate should be confident in her chances of winning the electoral votes in that state? Why or why not?

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Latest Statistics Questions

When a simple random poll of 1000 people says that a presidential candidate has of the support in
particular state, does this mean that the candidate should be confident in her chances of winning the
electoral votes in that state? Why or why not?