A newspaper sent a letter to 7500 persons in a town with the question "Do you agree with the decision to build a new football stadium in town?".
2100 people answered yes, 290 answered no and the rest didn't have any opinion. The newspaper took a random sample of 50 persons of the ones who didn't answer and it was showed that 22 said no, 8 said yes and 20 didn't know.
Calculate the percentage in favor of and against the decision

The right answer is:

In favor of: 31.9%
Against: 35.5%

I just need to know how to calculate it.

Just so it is clear, what is the amount of people who did not reply, as opposed to the amount of people who had no opinion on the matter.

Is this what you are being asked to work out? Or is the question to arrive at the percentages given?

The question still makes no sense. It won't matter if you post it in several more places.

Initial Result:

Yes: 2100/7500 = 0.28000
No: 290/7500 = 0.03867
No Opinion: (7500 - 2100 - 290)/7500 = 5110/7500 = 0.68133

Check: 0.28000 + 0.03867 + 0.68133 = 1.00000 -- Good!

The subsequent task is to do a better job on the "No Opinion" folks.

Yes: 5110*(8/50) = 817.6 ==> 818?
No: 5110*(22/50) = 2248.4 ==> 2248?
Still No Opinion: 5110 - 818 - 2248 = 2044

All told, then:

Yes: (2100+818)/7500 = 2918/7500 = 0.38907
No: (290+2248)/7500 = 2538/7500 = 0.33840
No Opinion: 2044/7500 = 0.27253

Check: 0.38907 + 0.33840 + 0.27253 = 1.00000 -- Good!

There are several technical problems with this:

1) It COMPLETELY ignores the "I know, but I'm not telling you" group. I'm pretty sensitive to this exclusion, since I believe firmly that pollsters have contributed to the destruction of American Politics.

2) It assumes the second round of questions extracts real opinions. Many who fail to respond the first time simply don't want to be bothered. When they are asked again, they give an answer just to get the pollster off their backs. This is not a good result.

3) Since my results don't match the "answer", I have suspicions about transcription. Did you switch the yes and no on the second round? I didn't check the arithmetic.

4) 7500 is a stunning number of letters. For such a letter, the standard thought used to be this: 2% is a good response - 150 returned. 4% is an excellent response - 300 returned. 6% is an astounding response - 450 returned. This survey has a 32% response rate? Give me a break!

You use the sample statistics for the second sample from the no opinions on the first sample to allocated all the no opinions in the first sample to the YES, NO and DON'T KNOW categories.

You have no opinions in first sample and we allocate to the NOs and to the YESs. Giving total YESs of , and NOs of . The problem is that these do not correspond to the percentages given in the answer.

This is assuming that "didn't have any opinion" on the first sample is the same as "did not answer" which are not normally the same thing (and if they did respond with no opinion it is statistical malpractice to do what is done here).

Without a statement of how the initial sampling frame was constructed this survey is invalid, as it is not usually posible to construct a proper random sample for a mail shot without a lot of careful work.

RonL

__________________
Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno