Section 5.6: Using the Sampling Distribution to Interpret Polls
Thus far, we have discussed the sample distribution of the true percentage of the population. Recall that we assumed that we knew the true value of the percentage p and then observed what the distribution of all possible samples of a given sample size would look like. However, we do not know the true percentage p. If we did, we wouldnβt need to take a sample at all!
The best estimate we can use for p is the sample proportion πΜ. We can use the identical Confidence Intervals and Margin of Error that we developed about p for πΜ.
So we are making a slight change of perspective now, but it is an important change. If you didnβt read the above paragraph carefully, you may think I am just repeating what we already did. The distinction to be made is that now, we are not assuming we know p. Instead we are assuming the sample proportion we obtain from a single simple random sample πΜ is the best estimate of p and that the distribution of πΜ behaves like the sampling distribution of p.
This result is stated below. Notice the only difference from our previous results is that p is replaced with πΜ.
Result: When we use a sample percentage πΜ to estimate the population percentage, the 95% Confidence Interval for the population using πΜ for sample size n is:
(πΜ – [latex]\frac{1}{\sqrt{n}}[/latex], πΜ + [latex]\frac{1}{\sqrt{n}}[/latex])
where MOE =Β [latex]\frac{1}{\sqrt{n}}[/latex] is a conservative estimate for the Margin of Error of the 95% Confidence Interval.
Now that we have extended the results of population percentages to sample percentages, we can interpret a sample percentage. We will do this almost identically to our previous work.
However, we will be given a single sample percentage.
Example 18
Worked Example β Finding the MOE and CI from a Sample Percentage
A pollster sampled n = 1156 people and found that favor of a proposition on the ballot in November. πΜ = 51% of likely voters sampled were in
A) Find the Margin of Error for this point estimate. (Round to two decimal points as needed)
Solution: We will use the conservative estimate for the Margin of Error, MOE = [latex]\frac{1}{\sqrt{n}}[/latex]
SinceΒ n = 1156, MOE = [latex]\frac{1}{\sqrt{1156}}[/latex] = [latex]\frac{1}{34}[/latex]
Written as a percentage rounded to two decimal places: [latex]\frac{1}{34}[/latex] β’ 100% β 2.94%
Answer: The Margin of Error is 2.94%.
B) Find the 95% Confidence Interval for this point estimate and interpret it.
Solution: The 95% Confidence Interval for the point estimate is (πΜ – [latex]\frac{1}{\sqrt{n}}[/latex], πΜ + [latex]\frac{1}{\sqrt{n}}[/latex])
πΜ = 51%
πΜ – [latex]\frac{1}{\sqrt{n}}[/latex] = 51% – 2.94% = 48.06%
πΜ + [latex]\frac{1}{\sqrt{n}}[/latex] = 51% + 2.94% = 53.94%
So the 95% Interval is (48.06%, 53.94%). The interpretation is below.
Answer: We are 95% confident that the true value of the percent of likely voters in favor of the Proposition lies between 48.06% and 53.94%.
Note: When we interpreted confidence intervals for the population proportion, we stated that 95 out of 100 samples are within the confidence interval. When we interpret the confidence interval for the single sample proportion we say we are 95% βconfidentβ that the true value lies in the confidence interval. We use the weaker term βconfidentβ because we were certain about the distribution of the true population distribution, but we are using an estimate when we use the single sample proportion.
Example 19
Worked Example β Comparing the MOE and CI from two Sample Percentages
An exit pollster sampled n = 1098 people as they left the voting place and found that 44% of the voters sampled were in favor of Candidate A and 56% of the voters sampled were in favor of Candidate B. (Round any results below to two decimal places as needed.)
A) Find the Margin of Error for this poll.
Solution: MOE = [latex]\frac{1}{\sqrt{n}}[/latex] β’ 100% = [latex]\frac{1}{\sqrt{1098}}[/latex] β’ 100% β 3.02%
Answer: The Margin of Error is 3.02%
B) Find the 95% Confidence Interval for the percentage of Candidate A’s votes and interpret it.
Solution: (πΜ – [latex]\frac{1}{\sqrt{n}}[/latex], πΜ + [latex]\frac{1}{\sqrt{n}}[/latex]) = (56% – 3.02%, 56% + 3.02%) = (52.98%, 59.02%)
Answer: We are 95% confident that the true value of the percent of Candidate A’s votes lies between 52.98% and 59.02%
C) Find the 95% Confidence Interval for the percentage of Candidate B’s votes and interpret it.
Solution: (πΜ – [latex]\frac{1}{\sqrt{n}}[/latex], πΜ + [latex]\frac{1}{\sqrt{n}}[/latex]) = (44% – 3.02%, 44% + 3.02%) = (40.98%, 47.02%)
Answer: We are 95% confident that the true value of the percent of Candidate B’s votes lies between 40.98%and 47.02%
D) Do the 95% Confidence Intervals for Candidate A and Candidate B overlap?
Solution: The largest value in Candidate Bβs interval is 47.02%and the smallest value in Candidate Aβs interval is 52.98%, so they do not overlap.
Answer: No. The confidence Intervals do not overlap.
E) Based on your previous answer, choose the appropriate interpretation:
Since the confidence intervals do not overlap, there is statistical evidence that the percentages are not equal. In addition, Candidate Bβs percentage is higher, so there is statistical evidence Candidate B is in the lead.
Answer: There is statistical evidence that Candidate B is in the lead.