State the poll question, the sample size n, and the margin of error (also known as sampling error or MoE) for the data.

Question

When the results of a survey or poll are published, the sample size and the margin of error should both be given. For example: 1,000 voters were surveyed, and 39% ± 2% of the voters agree with the president. In this example, N = 1,000, and the margin of error (MoE) is 2%.

NBC News Poll conducted by Hart Research Associates (D) and Public Opinion Strategies (R). Aug. 14-17, 2021. N=1,000 adults nationwide. Margin of error ± 3.1.
"When it comes to addressing climate change, do you believe that the federal government is doing enough to address climate change, not doing enough to address climate change, or is this an issue in which the federal government does not need to be involved?"
Doing	enough	Not doing	enough	Does not	need to be	involved	Unsure
%	%	%	%
8/14-17/21	14	52	31	3

Based on the poll above, answer the questions below.

1. State the poll question, the sample size n, and the margin of error (also known as sampling error or MoE) for the data.

2. State the confidence interval for the selected poll question using the MoE. What does this confidence interval estimate?

3. Use the Confidence Interval template below to calculate the confidence interval based on the sample size and the number of successes (the proportion you are interested in). Use a 95% level of confidence.

Sample Size for Proportion	Color Key:
Estimate of True Proportion	0.75	Cells that require student input
Sampling Error	0.03	Excel Intermediate Calculations
Confidence Level	95.00%	Excel Calculated Results
Major Headings
Z Score	-1.95996	Minor Headings
Reference/Check Points
Calculated Sample Size	801

4. State the following:

- What is the calculated confidence interval?

- How does this compare to the interval in #3 from the poll's margin of error?

5. Discuss potential biases that could skew sampling results.

6. Interpret the results of the poll selected using your own words and complete sentences. Note: Depending on the question asked, the poll may have more than one poll result — you only need to discuss one result.

Answer 1

Q1

The poll question appears in the problem as: When it comes to addressing climate change, do you believe that the federal government is doing enough to address climate change, not doing enough to address climate change, or is this an issue in which the federal government does not need to be involved?

The sample size (n) is given to be 1000 adults.

The margin of error (MoE) is given in the problem as +/- 3.1%.

Q2

The table tells us that 14% of the adults in the sample feel that the federal government is doing enough . The given MoE includes the confidence limits but we are not given what level of confidence so the best we may provide is a 100(1-α)% CI of 14 ± 3.1 or (10.9%, 17.1%). This interval estimates the true percent of the population of adults who feel that the federal government is sufficiently involvement in addressing climate change.

Q3

Using the given information we observe a sample proportion of 0.14 who feel the government is sufficiently involved in addressing climate change, 0.52 who feel the government should be more involved, 0.31 who feel the government should not be involved at all and 0.03 who are not sure how they feel on this issue. We can create 95% confidence intervals for all of these proportions (p-hat) directly by:

phat ± 1.96 * sqrt( phat * (1-phat) / n)

Percent of adults feeling that the government is doing enough to address climate change: 14% (95%CI 11.8% - 16.2%)

Percent of adults feeling that the government should be doing more to address climate change: 52% (95%CI 48.9% - 55.1%)

Percent of adults feeling that the government should not be involved in addressing climate change: 31% (95%CI 28.1% - 33.9%)

Percent of adults feeling unsure about the government's involvement in addressing climate change: 3% (95%CI 1.9% - 4.1%)

Q4

Response Category Percent Calculated 95% CI 95% CI using MoE

Doing Enough 14 (11.8 - 16.2) (10.9 - 17.1)

Not Doing Enough 52 (48.9 - 55.1) (48.9 - 55.1)

Should Not Be Inv. 31 (28.1 - 33.9) (27.9 - 34.1)

Unsure 3 ( 1.9 - 4.1) ( -0.1 - 6.1)^*

^*Because it is not possible to have fewer than zero responses this interval could be truncated to (0 - 6.1) without loss of confidence.

The sampling error for a proportion is sqrt( phat * (1-phat) / n). This function is maximized at phat=0.5. The MoE is calculated using 0.5 so when the sample proportion is close to 0.5 we may expect to see calculated confidence intervals of approximately the same width as the MoE interval. We may observe this directly in the "Not Doing Enough" row where the sample proportion is 0.52 and the calculated and MoE intervals are essentially identical. As the percentages get closer to zero we see the MoE interval becoming much wider compared to the interval we observe through calculation.

Q5

There are a number of potential sources of bias that could be leading to skewed results. I provide a short listing here but there are probably many other good answers as well:

1. The question is worded quite vaguely and "doing enough" may mean different things to different people.
2. The topic of climate change is also a politically charged issue that may be leading respondents to answer based on how they feel about climate change and not about how they perceive the government's involvement
3. The sample of adults may not be representative of the population. This could happen due to non-response where people in certain demographic groups feel less comfortable voicing opinions about the government. This could also happen if the sampling frame inadvertently leaves out certain demographic groups. If this was an internet survey where people had to follow a link placed on an ad or at the end of an article then the respondents could be from a very specific sector of the population.
4. All we know is that we have 1000 respondents who answered this question. We do not know how many people were sampled to arrive at this number of completes. If the survey was sent out to 100,000 adults and only 1000 of them were returned with this question completed then we should be careful in suggesting that these data are indicative of all adults. This is also a concern if this was an internet survey attached to an article or ad link as we do not know how many people had access to the survey and chose not to respond.
5. The survey item regarding government involvement and climate change is presented as an isolated item but the question could have appeared anywhere in the survey. We have no information on what questions preceded this one in the survey or where the question appeared in the survey instrument. If this question was asked first then we might assume there was less opportunity to bias the respondents' answers. If the question appeared after an awareness list of government activities to address climate change or a list of failed endeavors for addressing climate change then the respondents' opinions may be effectively swayed. The survey was also conducted at the start of hurricane season in the US which could have led respondents to answer questions about climate change in a different way than what might be observed in different parts of the year.

Q6

(For this section I will focus on the percent of adults indicating that the government is doing enough to address climate change)

Our survey indicates that an estimated 14% (95% CI 11.8% - 16.2%) of adults in the United States who had access to this survey and were eligible to participate believe the government is doing enough to address climate change. While the true percentage of adults agreeing that the government's efforts are sufficient is unknown, we can be very confident that the true percentage is somewhere between 11.8% and 16.2%. This is because our sample of 1000 people was selected using a random process and, if we were to repeat it many times, we would expect to produce different estimates and different intervals each time. These intervals are calculated in such a way that about 95% of all the intervals we might observe contain the true percentage. Therefore, we trust that the interval we observe here is one of those correct intervals and we then may be 95% confident that the true percentage is somewhere between 11.8% and 16.2%. This percentage does not apply to adults residing in the US who either did not have access to the survey or were ineligible to participate for whatever reason.