7) President’s job approval rating is frequently reported in news programs. You will hear statements such as: “Among likely voters, the President’s job approval rating is 46% with a margin of error of 6%.”
a) What percent confidence interval do you think they are reporting?
b) How do you explain the 6% margin of error to someone who is not mathematically trained? In other words, how do you explain it in plain English?
c) Someone claims that although the confidence interval is , the probability that the president is favored by the majority of likely voters is low because 50% to 52% portion is only 1/6 of the whole interval. Do you agree with this statement? Explain!
a) Based on the information given, they are likely reporting a 95% confidence interval. This is a common level of confidence used when reporting survey results.
b) The margin of error of 6% means that there is a range of plus or minus 6 percentage points around the reported approval rating. In other words, if the President's job approval rating is reported as 46%, it means that the actual rating could be as high as 52% or as low as 40%, with 46% being the most likely value. The margin of error is a measure of the uncertainty in the survey results and accounts for sampling variability.
To explain it in plain English, imagine that you are trying to estimate how many people in a city like a particular restaurant. Instead of asking every single person in the city, you randomly select a sample of people and ask them if they like the restaurant. The margin of error takes into account that the opinions of the entire population might be slightly different from the opinions of the sampled individuals. So, the margin of error tells us how much the reported approval rating could vary from the true rating in the entire population of likely voters.
c) No, I do not agree with that statement. The probability that the president is favored by the majority of likely voters is not directly interpreted from the confidence interval. The confidence interval simply tells us the range of values within which the true approval rating is likely to fall. It does not tell us anything about the probability of any specific value within that interval.
In this case, the 50% to 52% portion being only 1/6 of the whole interval does not imply that the probability of the president being favored by the majority of likely voters is low. It simply means that the range of values above 50% is smaller than the range below 50%. However, it is still possible for the true approval rating to be above 50% within that range. The confidence interval provides information about the range of uncertainty, not about the probability of specific values occurring within that range.