Chapter 11.5, Problem 25E

(a)

To determine

To find: thenumbers of voters were surveyed.

Answer to Problem 25E

There are 500 voters in the survey.

Explanation of Solution

Given:

235voters are 47% of the voters in the survey

Calculation:

Let the total number of voters be n.

Given that 235voters are 47% of the voters in the survey. That is,

  $\begin{array}{l}\begin{array}{l}47\%\text{ of }n=235\\ 0.47n=235\end{array}\hfill \\ n=500\hfill \end{array}$

So, $n=500$ or there are 500 voters in the survey.

Conclusion:

Therefore, there are 500 voters in the survey.

(b)

To determine

To find: the margin of error for the survey.

Answer to Problem 25E

The margin of error is about 4.5%.

Explanation of Solution

Calculation:

Use the margin of error formula,

  $\begin{array}{l}\text{Margin of error }=\pm \frac{1}{\sqrt{n}}\\ =\pm \frac{1}{\sqrt{500}}\\ =\pm 0.045\end{array}$

Thus, the margin of error is about 4.5%.

Conclusion:

Therefore, the margin of error is about 4.5%.

(c)

To determine

To find: an interval of the exact percent of all voters

Answer to Problem 25E

An interval of the exact percent of all voters for candidate B is between 48.5% and 57.5%

An interval of the exact percent of all voters for candidate A is between 42.5% and 51.5%.

Explanation of Solution

Calculation:

If p is the percent of the sample responding in a certain way, andis the margin oferror, then the percent of the population who would respond the same way is likely to be between $p-\frac{1}{\sqrt{n}}$ and $p+\frac{1}{\sqrt{n}}$ .

Now, 47% of the sample voted for candidate A and the remaining, that is 53% voted for candidate B.

To find the interval that is likely to contain the exact percent of all voters who voted for candidate A, subtract and add 4.5% to the percent of voters surveyed who voted for candidate A (47%).

  $p-\frac{1}{\sqrt{n}}=47\%-4.5\%=42.5\%$

and

  $p+\frac{1}{\sqrt{n}}=47\%+4.5\%=51.5\%$

Thus, it is likely that the exact percent of all voters who voted for candidate A is between 42.5% and 51.5%.

And

To find the interval that is likely to contain the exact percent of all voters who voted for candidate B, subtract and add 4.5% to the percent of voters surveyed who voted for candidate B (53%).

  $p-\frac{1}{\sqrt{n}}=53\%-4.5\%=48.5\%$

and

  $p+\frac{1}{\sqrt{n}}=53\%+4.5\%=57.5\%$

Thus, it is likely that the exact percent of all voters who voted for candidate B is between 48.5% and 57.5%

Conclusion:

Therefore, an interval of the exact percent of all voters for candidate B is between 48.5% and 57.5%

An interval of the exact percent of all voters for candidate A is between 42.5% and 51.5%.

(d)

To determine

To find: whether the candidate B won or not. Find the number of votes candidate needs to win if he won’t win.

Answer to Problem 25E

Candidate B needs 273 voters to be confident for a win.

Explanation of Solution

Calculation:

Based on the above intervals, it is cannot be confident about the winner. Because for the interval that is likely to contain the exact percent of all voters who voted for the candidates overlaps.

Now, you have to find the number of people in the sample that need to vote for candidate B. SO that you can be confident that candidate B will win. For that, you need to find the least number of voters for candidate B so that the intervals do not overlap

Let the percent of voters in the sample that vote for candidate B is p. Then percent of voters in the sample that vote for candidate B is $p_1=\left(100-p\right)$ . So, for the intervals to not to overlap, you need to have that smallest number of interval for candidate B is greater than the largest number of interval for candidate A, that is

  $\begin{array}{l}{p}_1+4.5<p-4.5\\ \left(100-p\right)+4.5<p-4.5\\ 100+4.5+4.5<p+p\\ 109<2p\\ 54.5<p\end{array}$

So, at least 54.6% of voters in the sample should vote for candidate B, so that you can be confident that candidate B will win.

Thus, the number of people in the sample that need to vote for candidate B, so that you can be confident that candidate B will win is,

  $54.6\%\text{ of }500=\frac{54.6}{100}\times 500=273$

Candidate B needs 273 voters to be confident for a win.

Conclusion:

Therefore, Candidate B needs 273 voters to be confident for a win.