29. problem solving a survey reported that 47% of the voters surveyed, or about 235 voters, said they voted…

29. 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.\na. how many voters were surveyed?\n1 voters\nb. what is the margin of error for the survey? round your answer to the nearest tenth.\nabout ± 2 %\nc. 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.\ncandidate a: between 3 % and 4 %\ncandidate b: between 5 % and 6 %\ncorrect answers:\nd. based on your intervals in part (c), can you be confident that candidate b won?\nyes\nno\nhow many people in the sample would need to vote for candidate b for you to be confident that candidate b won?

29. 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.\na. how many voters were surveyed?\n1 voters\nb. what is the margin of error for the survey? round your answer to the nearest tenth.\nabout ± 2 %\nc. 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.\ncandidate a: between 3 % and 4 %\ncandidate b: between 5 % and 6 %\ncorrect answers:\nd. based on your intervals in part (c), can you be confident that candidate b won?\nyes\nno\nhow many people in the sample would need to vote for candidate b for you to be confident that candidate b won?

Answer

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