1. ## Hypothesis Testing

Suppose that we survey $500$ randomly selected Country $X$ voters about whether or not they think Mr.A should leave the race, and $254$ of the $500$ voters say "yes". Test the hypothesis that greater than $50$% of Country $X$ voters believe that Mr.A should quit the race. Find the $p$-value for this test. What does the value indicate, or mean?

How do I do this hypothesis testing problem? Help would be much appreciated.

2. ## Re: Hypothesis Testing

The distribution is a binomial one. Assume that the null hypothesis is true. So, the p-value is the probability that 254 or more of the voters would say "yes".

That is given by:

$\displaystyle \displaystyle \left(\dfrac{1}{2}\right)^{500}\sum_{n=254}^{500} \dbinom{500}{n}$

But, this value tells you nothing. No level of significance was provided for us to either accept or reject the null hypothesis.

3. ## Re: Hypothesis Testing

Thank you for the reply. So the p value is essentially the cumulative binomial distribution up till $\displaystyle P(500\ge X\ge254)$? So the p-value means nothing because we have no significance value to compare it to? I get $\displaystyle P(500\ge X\ge254)=0.65631834595981$, is this the p-value?, just double checking.

Another question, why is it from 254 to 500? not 0 to 254?

4. ## Re: Hypothesis Testing

Originally Posted by gaussrelatz
Thank you for the reply. So the p value is essentially the cumulative binomial distribution up till $\displaystyle P(500\ge X\ge254)$? So the p-value means nothing because we have no significance value to compare it to? I get $\displaystyle P(500\ge X\ge254)=0.65631834595981$, is this the p-value?, just double checking.

Another question, why is it from 254 to 500? not 0 to 254?
Correct (although I was using the complement, $\displaystyle P(X\ge 254) \approx 0.377139$ while you were using $\displaystyle P(X\le 254)\approx 0.65632$. If this is for a course, I am not sure what the instructor is looking for. Perhaps that it is unlikely that you would consider rejecting the null-hypothesis based on this result (whichever value you choose for the p-value is unlikely to invalidate the null-hypothesis).

A Bayesian Hypothesis Test may be in order. Knowing that the value you got from experimentation was 254/500, what is the probability that the null-hypothesis is correct?

That could be:

$\displaystyle \displaystyle \sum_{n=250}^{500}\dbinom{500}{n}\left(\dfrac{254} {500}\right)^n\left(\dfrac{246}{500}\right)^{500-n} \approx 0.656410918498859995795195761822985013307551984900 928813437$

This is the probability that if you find a sample of 500 people has 254 respondents say "yes" that the statement (at least 50% of the population of Country X would say "yes") assuming that the sample is truly random and representative.

5. ## Re: Hypothesis Testing

Actually this is a problem that one of my friends asked me, as he is stat major, and Im a pure math major, I like briefly learned this in high school so I forgot most of it thereby I asked the problem on the forum as it seemed quite unusual. As all the problems I looked in the internet seemed to give some $\displaystyle \alpha$ for comparison. But this problem didnt. I am not sure what the Bayesian hypothesis test is, not familiar with it at all.

6. ## Re: Hypothesis Testing

Originally Posted by gaussrelatz
Actually this is a problem that one of my friends asked me, as he is stat major, and Im a pure math major, I like briefly learned this in high school so I forgot most of it thereby I asked the problem on the forum as it seemed quite unusual. As all the problems I looked in the internet seemed to give some $\displaystyle \alpha$ for comparison. But this problem didnt. I am not sure what the Bayesian hypothesis test is, not familiar with it at all.
The idea of Bayesian hypothesis testing is, rather than assume a null-hypothesis, you accept that the experimental results is all that you know. Then, knowing you found those results, what is the probability that the hypothesis is true?

Bayesian hypothesis testing can become far more complicated than this. I gave a very rudimentary example of how the analysis may be conducted.

7. ## Re: Hypothesis Testing

Originally Posted by SlipEternal
The idea of Bayesian hypothesis testing is, rather than assume a null-hypothesis, you accept that the experimental results is all that you know. Then, knowing you found those results, what is the probability that the hypothesis is true?

Bayesian hypothesis testing can become far more complicated than this. I gave a very rudimentary example of how the analysis may be conducted.
So my friend came up with the solution by his professor and he actually got the s.d and variance by letting the sample mean as 254/500 and then finding the t-value then comparing it with the table to come up with the p-value. I dont understand this method at all, could you possibly do it in this way? If so exactly how would i do that?