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 37.9 % and 44.1 %
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 1020 people in the U.S., 41 % reported that their top priority for saving is retirement. Therefore, we have that n= 1020. Let's find the margin of error by substituting this value in the above expression.
Margin of Error: ± 1/sqrt(1020) ≈ ± 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 about 0.031. Now we can use that value to find the interval, which should contain the exact percent of all U.S. adults whose top priority for saving is retirement. 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 41 % of the surveyed people reported that their top priority for saving is retirement. To find the lower boundary of the interval, we will subtract 0.031 from p= 0.41. Similarly, to find the upper boundary we will add 0.031 to p= 0.41. Let's start by finding the lower boundary.
The boundaries tell us the interval in which the actual percent of the population would report that their top priority for saving is retirement.
Between 37.9 % and 44.1 %
