a The margin of error gives a limit on how much the responses of a sample would differ from the responses of the population.
B
b Use the margin of error to find the desired interval.
A
a About 3.1 %
B
b Between 72.9 % and 79.1 %
Practice makes perfect
a We want to find the margin of error for the sample data. A margin of error gives a limit on how much the responses of a sample would differ from the responses of the population. If we let n be the size of a random sample taken from a large population, we can write an expression for the margin of error.
Margin of Error: ± 1/sqrt(n)
We are told that in a survey of 1022 people in the U.S., 76 % said that more emphasis should be placed on producing domestic energy from solar power. Therefore, we have that n= 1022. Let's find the margin of error by substituting this value in the above expression.
Margin of Error: ± 1/sqrt(1022) ≈ ± 0.031
The margin of error for the survey is about ± 0.031 or ± 3.1 %.
b In Part A we calculated the margin of error is 0.031. Now we can use that value to find the interval, which should contain the exact percent of all U.S. adults who said that more emphasis should be placed on producing domestic energy from solar power. Let p be the percentage of the sample responding a certain way, written as a decimal.
Interval
Between p- 1sqrt(n) and p+ 1sqrt(n)
We are told that 76 % of the surveyed people said that more emphasis should be placed on producing domestic energy from solar power. To find the lower boundary of the interval, we will subtract 0.031 from p= 0.76. Similarly, to find the upper boundary we will add 0.031 to p= 0.76. Let's start by finding the lower boundary.
The boundaries tell us the interval in which the actual percent of the population would think that that more emphasis should be placed on producing domestic energy from solar power.
Between 72.9 % and 79.1 %
