Add and subtract the margin of error from each of the percentage values.
D
d
In the worst scenario, what is the support for Candidate B? Is it high enough for them to win?
A
a
500
B
b
About ± 4.5 %
C
c
Candidate A: Between 42.5 % and 51.5 % Candidate B: Bbetween 48.5 % and 57.5 %
D
d
Answer: No. Required Number of People: 273
Practice makes perfect
a
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. Our goal is to find the total number of people surveyed.
Candidate A
Candidate B
Support
47 %
100 - 47 = 53 %
Let n be the number of people surveyed. We were told that 47 %=0.47 of those people, which is approximately 235 people, voted for Candidate A. Let's combine these two facts to form an equation.
0.47n = 235
We can now solve the equation for n.
We want to find the margin of error for the survey. For that, recall that when a random sample of size n is taken from a large population, the margin of error can be approximated with the following formula.
Margin of error = ± 1/sqrt(n)
Notice that we have already found the size of the sample in Part A. Therefore, let's substitute n=500 into the formula, then evaluate.
We are asked to find the interval that is likely to contain the exact percent of all voters who voted for each candidate. First, let's recall the results of the survey.
Candidate A
Candidate B
Support
47 %
53 %
Now, if the percent of the sample responding a certain way is p, then the percent of the population who would respond the same way is likely to be less than the margin of error from p. Therefore, it is likely to be between the two following values.
p - 1/sqrt(n) and p + 1/sqrt(n)
The support for Candidate A is 47 %. In order to find the interval, we need to add and subtract the margin of error found in Part B from the percentage of voters who chose Candidate A.
47 % - 4.5 % &= 42.5 %
47 % + 4.5 % &= 51.5 %
Now, the support for Candidate B is 53 %. Following similar steps, let's add and subtract the margin of error. This will give us the interval that is likely to contain the exact percent of all voters who voted for Candidate B.
53 % - 4.5 % &= 48.5 %
53 % + 4.5 % &= 57.5 %
d
We want to decide if we can be confident about Candidate B winning. For that, let's take a look at our findings from Part C.
Candidate A
Candidate B
Support
47 %
53 %
Lower margin
42.5 %
48.5 %
Upper margin
51.5 %
57.5 %
Looking at the results, we see that we cannot be confident in Candidate B winning. Taking into consideration the margin of error for the survey, the likely support for Candidate B could be as low as 48.5 %, which is less than the required 50 % to win.
48.5 % < 50 %
Moving on, let p be the percentage of Candidate B's supporters needed for Candidate B to win. In order to be confident that Candidate B won, the lower margin, which is the support minus the margin error, has to be at least 50 %.
p- 4.5 % = 50 %
Let's solve the resulting equation for p.
p-4.5 %= 50 % ⇔ p = 54.5 %
The required percentage of voters is 54.5 % = 0.545. Knowing that a total of 500 people participated in the survey, let's calculate the number of the people who need to vote for Candidate B so that we can be sure that Candidate B won.
0.545( 500) &= 272.5
&≈ 273
The number of people required is 273.
