I have just started a course in statistics and have some general questions that have arisen trying to solve the following question:

A survey organisation wants to take a simple random sample in order to estimate the percentage of people who have seen a certain programme. The sample is to be as small as possible. The estimate is specified to be within 1 percentage point of the true value; i.e., the width of the interval centered on the sample proportion who watched the programme should be 1%. The population from which the sample is to be taken is very large. Past experience suggests the population percentage to be in the range 20% to 40%. What size sample should be taken?

I think I have to use this and solve for n

$$1.96\sqrt{\frac{\pi (1-\pi )}{n}}=.01$$

where $\pi $ is the sample estimated proportion of people who watch the programme.

Now does this mean that I am 95% sure that I am within 1% accuracy? I also am not aware as to how I can find π though I have read I could use the population standard deviation instead and suspect I would have to use that as I am given some information- that the pop proportion is 20%-40%.

Finally in general what is being said here:

$$\pi \pm 1.96\sqrt{\frac{\pi (1-\pi )}{n}}=.01$$

My notes at the moment just say it contains the population mean 95% of the time.... why? I think if I had some graphical understanding of what was going on everything would be much simpler for me.

A survey organisation wants to take a simple random sample in order to estimate the percentage of people who have seen a certain programme. The sample is to be as small as possible. The estimate is specified to be within 1 percentage point of the true value; i.e., the width of the interval centered on the sample proportion who watched the programme should be 1%. The population from which the sample is to be taken is very large. Past experience suggests the population percentage to be in the range 20% to 40%. What size sample should be taken?

I think I have to use this and solve for n

$$1.96\sqrt{\frac{\pi (1-\pi )}{n}}=.01$$

where $\pi $ is the sample estimated proportion of people who watch the programme.

Now does this mean that I am 95% sure that I am within 1% accuracy? I also am not aware as to how I can find π though I have read I could use the population standard deviation instead and suspect I would have to use that as I am given some information- that the pop proportion is 20%-40%.

Finally in general what is being said here:

$$\pi \pm 1.96\sqrt{\frac{\pi (1-\pi )}{n}}=.01$$

My notes at the moment just say it contains the population mean 95% of the time.... why? I think if I had some graphical understanding of what was going on everything would be much simpler for me.