A researcher reports survey results by stating that the standard error of the mean is $20 .$ The population standard deviation is 500 .
a. How large was the sample used in this survey?
b. What is the probability that the point estimate was within $\pm 25$ of the population mean?
all right. This question gives us the standard air of the mean and the population standard deviation, and party wants us to calculate this sample size. Find any so we can do this by an equation that relates it's and toe standard error and Population Sigma. So that's just our formula for standard error, where the standard air of X bar equals the population standard deviation over the square to the sample size. So then we can plug in the values we know 20 equals 500 over square root of our sample size or 20 squared of n equals 500 then dividing by 20 on each side, we get square root of men equals 25 and then squaring both sides. We find that and equals 625. So this was quite an interesting question because we had to use just a bit of algebra to solve this equation for n which we usually no end and B wants to know. The probability that we're within 25 of the meeting, which is the same thing, is asking What is the probability that we get a next bar between you minus 25 mu plus 25 but it doesn't give us mu. But that actually doesn't matter because when dealing with normal curves, the location of the mean doesn't matter. As long as we know the standard air, all the probabilities will be the same if if Muse one negative three or 1000 doesn't matter, because we know the standard error and it's not affected by the mean. So just to make things easier, let let me you equal zero. So now we can do normal. CDF of our lower bound is negative, 25 are upper bound is positive 25 Our mean that we picked zero and our standard error was given to us as 20. And this works out to be 0.7887 And just to prove that just to prove that this equation works, regardless of what value of mu, you can try to substitute whatever you want. Try it with musicals 30 and you'll see that this still gives the same answer. If you're not