Question

1. problem solving a survey reported that 47% of the voters surveyed, or about 235 voters, said they voted for candidate a and the remainder said they voted for candidate b.

a. how many voters were surveyed?
1 voters
b. what is the margin of error for the survey? round your answer to the nearest tenth.
about ± 2 %
c. for each candidate, find an interval that is likely to contain the exact percent of all voters who voted for the candidate. round your answers to the nearest tenth.
candidate a: between 3 % and 4 %
candidate b: between 5 % and 6 %
correct answers:
d. based on your intervals in part (c), can you be confident that candidate b won?
yes
no
how many people in the sample would need to vote for candidate b for you to be confident that candidate b won?

Explanation:

Step1: Find total number of voters surveyed

Let the total number of voters surveyed be $x$. We know that $47\%$ of $x$ is 235. So, $0.47x = 235$. Solving for $x$, we get $x=\frac{235}{0.47}=500$.

Step2: Calculate margin of error

The formula for margin of error for a sample size $n = 500$ (assuming a 95% confidence - level) is $E=\frac{1}{\sqrt{n}}$. Substituting $n = 500$, we have $E=\frac{1}{\sqrt{500}}\approx\frac{1}{22.36}\approx0.045$ or $4.5\%$.

Step3: Find intervals for candidates

For Candidate A, the sample percentage is $47\%$. The lower - bound of the interval is $47 - 4.5=42.5\%$ and the upper - bound is $47 + 4.5 = 51.5\%$.
For Candidate B, the sample percentage is $100 - 47=53\%$. The lower - bound of the interval is $53 - 4.5 = 48.5\%$ and the upper - bound is $53+4.5 = 57.5\%$.

Step4: Determine if Candidate B won

Since the interval for Candidate A is $42.5\%$ to $51.5\%$ and the interval for Candidate B is $48.5\%$ to $57.5\%$, there is an overlap between the two intervals. So, we cannot be confident that Candidate B won.

Answer:

1. 500 voters
2. about $\pm4.5\%$
3. Candidate A: between $42.5\%$ and $51.5\%$
4. Candidate B: between $48.5\%$ and $57.5\%$
5. No