Chapter 6.4, Problem 31PS

Solution

(a)

To find: It needs to be determined the percent of voters in the sample voted for candidate A and for candidate B.

The percent of voters in the sample voted for candidate A is 47% and for candidate B is 53%.

Given information: A survey reported that 235 out of 500 voters in a sample voted for candidate A and the remainder voted for candidate B.

Calculation: The percent of voters who voted for candidate A is as:

  $\frac{235}{500}\times 100=47\%$

The percent of voters who voted for candidate B is as:

  $100-47\%=53\%$

Hence the percent of voters in the sample voted for candidate A is 47% and for candidate B is 53%.

(b)

To find: It needs to be determined the margin of error for the survey.

The margin of error for the survey is $\pm 4.5\%$ .

Given information: A survey reported that 235 out of 500 voters in a sample voted for candidate A and the remainder voted for candidate B.

Formula used: The formula that is used for evaluating the margin of error is $\pm \frac{1}{\sqrt{n}}$ .

Calculation: The number of students is 500 so one has $n=500$, thus the margin of error is evaluated as:

  $\begin{array}{c}\pm E=\pm \frac{1}{\sqrt{n}}\\ =\pm \frac{1}{\sqrt{500}}\\ =\pm 0.045\\ =\pm 4.5\%\end{array}$

Hence the margin of error for the survey is $\pm 4.5\%$ .

(c)

To find: It needs to be determined an interval that is likely to contain the exact percent of all the voters who voted for the candidate.

The interval for candidate A will be $\pm 42.5\%$ to $\pm 51.5\%$ and for candidate B will be $\pm 48.5\%$ to $\pm 57.5\%$ .

Given information: A survey reported that 235 out of 500 voters in a sample voted for candidate A and the remainder voted for candidate B.

Explanation: For finding interval for candidate A the margin of error 4.5% is subtracted and added to percent of all candidates that voted for candidate A.

  $47\%\pm 4.5\%=42.5\%\text{ to 51}\text{.5\%}$

For finding interval for candidate B the margin of error 4.5% is subtracted and added to percent of all candidates that voted for candidate A.

  $53\%\pm 4.5\%=48.5\%\text{ to 57}\text{.5\%}$

Hence the interval for candidate A will be $\pm 42.5\%$ to $\pm 51.5\%$ and for candidate B will be $\pm 48.5\%$ to $\pm 57.5\%$ .

(d)

To find: It needs to be determined whether candidate B will won and if not then how many people in the sample would need to vote for candidate B.

The candidate B will not necessarily win as the intervals overlap and for candidate B to won at least 273 votes are needed.

Given information: A survey reported that 235 out of 500 voters in a sample voted for candidate A and the remainder voted for candidate B.

Explanation: The candidate B will not necessarily win as the intervals overlap and for the intervals not to overlap one needs at least 9% difference between the confidence intervals, so one has 47% for candidate A and 53% for candidate B which means 6% difference and for making the remaining 3% one needs to increase the percentage of voters for candidate B by 1.5% and decrease the percentage of voters for candidate A by 1.5%.

Thus, needed percentage of voters for candidate B is as:

  $1.5\%+53\%=54.5\%$

The required voters is evaluated as:

  $\frac{54.5}{100}\times 500=272.5\approx 273$

Hence the candidate B will not necessarily win as the intervals overlap and for candidate B to won at least 273 votes are needed.