   SEARCH HOME Math Central Quandaries & Queries  Question from Fara, a student: It is common for public opinion polls to have a " confidence level" of 95%, meaning that there is a 0.95 probability that the poll results are accurate within the claimed margins of error. If six different organizations conduct independent polls, what is the probability that all six of them are accurate within the claimed margins of error? Does the results suggest that with a confidence level of 95%, we can expect that almost all polls will be within the claimed margin of error? Hi Fara,

It is common that public opinion polls are often reported with a confidence level of 95% but this DOES NOT MEAN that there is a 0.95 probability that the poll results are accurate within the claimed margins of error. This is a complete misinterpretation of what confidence intervals are. Once you have the poll results then they are either accurate within the margin of error or they are not. There is no probability involved at all.

When constructing a public opinion poll with a certain margin of error the procedure is designed so when the poll is conducted there is a 95% probability that the resulting confidence interval will "capture" the true value. That is that the true value will be within the margin of error. Once the poll is conducted there is no longer any probability involved. You either captured the true value or you didn't. The probability is in the procedure not the result.

I sometimes explain this to my students with a dart-board example. I am going to throw a dart at the board and try to hit the bulls eye. This is not a standard dart however, in place of the point on the end is a suction cup. I succeed in hitting the bulls eye if the suction cup covers it. Hence I have a margin of error which is the radius of the suction cup. With practice you can get good at this, perhaps good enough that 95% of the time you capture the bulls eye under the suction cup. There are 6 of us who have achieved this level of skill so you might ask if all six of us independently throw a dart, what is the probability that we all succeed in covering the bulls eye? The answer is 0.95 × 0.95 × ··· × 0.95 = 0.956.

I hope this helps,
Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.