Stats challenge question

Hey,

I'd really appreciate any help on this one:

(a) A fair coin is flipped N times. What is the probability of all the results being heads?

[I got (1/2)^N]

(b) How many consecutive heads would you have to observe before concluding that the coin is actually biased?

The hint says to use a 2-tailed test at the 5% significance level, but you can't use a calculator or statistical tables

Thanks

First one is obviously correct.

Second one, I presume you mean 95% significance? Could you estimate the mean and std dev and hence estimate +/- 2 std dev (1.96 would probably be more accurate) from the mean?

I'm not sure how I would work out the mean or standard deviation from the data given?

Why not have a think about it for a few minutes. You flip a coin N times, what is the mean number of heads and the std dev?

So would the mean be (N/2)/N = 1/2 ??

Which would make the std dev √(((N/2)^2)/N-(1/2)^2) = √((N-1)/4)

Which doesn't sound right

looks about right, maybe N/4 in the last one. What would the 95% intervals be as a function of N

Thanks

Would it be something along the lines of 2 √((N)/4) + 1/2 ?

Yes, mean +/- 2 std devs is approximately 95% confidence intervals. Are you ok now?

So would the mean be (N/2)/N = 1/2 ??

Which would make the std dev √(((N/2)^2)/N-(1/2)^2) = √((N-1)/4)

Which doesn't sound right

looks about right, maybe N/4 in the last one. What would the 95% intervals be as a function of N

Forgive my ignorance but shouldn't mean be N/2?

Yes, I didn't correct that but did notice it :-). Mean probabilty is 1/2, mean number of occurrences is N/2.

Forgive my ignorance but shouldn't mean be N/2?

I'm not sure

I thought the number of times N would come up was N/2 (on average?) and the total number of tests was N, hence (N/2)/N, but I'm not 100% on that first bit

Yes I agree with you. The number of time heads would come up would be N/2 and the total number of trials was N so mean (expected) probability would be 1/2 as you said.

It wasn't clear to me which one you were referring to, hence my question. Sorry!

(b) How many consecutive heads would you have to observe before concluding that the coin is actually biased?

The hint says to use a 2-tailed test at the 5% significance level, but you can't use a calculator or statistical tables

Not wishing to muddy the waters, but I interpret this rather differently.

We're not interested in the number of heads out of N tosses.

We're interested in the number of consecutive heads.

So, we're simply looking for the least N, such that (1/2)^N < 0.025.

I'll admit that the wording of the question isn't entirely consistent with that, in that the hint talks about a 2-tailed test.

Can you provide a link/scan of the original question?

The precise wording of the question verbatim is:

(a) A fair coin is flipped N times. What is the probability of all of the results being heads?
(b) You become suspicious. How many consecutive heads would you have to observe before concluding that the coin is, in fact, biased?
[Hint: Use a 2-tailed test at the 5% significance level.]

Not wishing to muddy the waters, but I interpret this rather differently.

We're not interested in the number of heads out of N tosses.

We're interested in the number of consecutive heads.

So, we're simply looking for the least N, such that (1/2)^N < 0.025.

I'll admit that the wording of the question isn't entirely consistent with that, in that the hint talks about a 2-tailed test.

Can you provide a link/scan of the original question?

Question seems "off" to me.