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The question nearly covers it all. I clearly have a binomial distribution, however the variance and standarddeviations are unknown, the population is also just n = 50. How should I approximate this? Do I have to do a normal approximation with the Normal-distribution or can I use the Student's t-distribution? I am persuaded to use the t-distribution since the std. is unknown, the relatively small population is also a factor. If I can't use Student's t-distribution, please enlighten me. In my mind this is like approximating the binomial distribution by aprroximating the normal distribution with the t-distribution.

Further information:

I read that when the H0 hypothesis is true the variance is known as Var(X) = np(1 − p). However in my situation the H0 hypothesis is false, both when using a Z-test and a T-Test.

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    $\begingroup$ What exactly do you need the approximation for? You mention an H0, but you don't say what it is, and why you want to test it. $\endgroup$ – Lewian Feb 24 at 11:51
  • $\begingroup$ H0: p0 = 1/8, H1: p0 > 1/8. Result p_test = 11/50 = 0.22, alpha = 0.05 => z_alpha = 1.645, t_alpha,ν ≈ 1.676. (Had to use 50 degrees of freedom, table did not have 49) Test observer, Z = 2.03 (or test observer, T = 2.03, don't know which one is correct) Either way => Discard H0. The test is done to test the winning chances of a game, and argue whether or not an error within the code of a game written in python exists. When discarding H0 errors in the code seems probable. When staying with H0 the code seems to work as intended. $\endgroup$ – JMy Feb 24 at 12:11
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    $\begingroup$ You can test using the binomial distribution directly, no approximation needed. binom.test in R gives p=0.05233, a borderline result. I'd say H0 is somewhat in doubt but the evidence against it is not crystal clear. $\endgroup$ – Lewian Feb 24 at 12:25
  • $\begingroup$ That's definitely borderline. This is however a much better result, which I would prefer in the real world! This question however is derived from a task within an exam I took. Where normal approximation is the main part of the problem. What do you think about the Z/T-test issue with that in mind? $\endgroup$ – JMy Feb 24 at 12:33
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You have a binomial distribution $B(n,p)$, where $n$ is the number of trials, and $p$ is the probability of success. For the binomial distribution, the random variable is $K$, the number of successes, and it has expected value $\mu = np$ and variance $\sigma^2 = np(1-p)$. In your example, $n = 50$, and $p$ is unknown. Let $k$ be the number of successes that you observed in those $n$ trials, and let $\hat{p} = k/n$ be your estimated probability of success.

Your last paragraph about the truth of the null hypothesis is a common misunderstanding, let me try and explain. An hypothesis test is similar to a proof by contradiction: First you make an assumption, then you prove that this assumption cannot be true, and hence you conclude that the opposite must be true.

In hypothesis testing, you first make an assumption (the null hypothesis, $H_0$), then you provide evidence (your observations), and if that evidence strongly contradicts your first assumption you state that there is evidence for the opposite. Pay attention to the word "evidence" here, signifying that you will not have definite proofs and you will need to make a choice of what is good enough evidence through a predefine threshold for the $p$-value. Usually this is a standard specific to your scientific community, often $\alpha = 0.05$ or $\alpha = 0.01$, but you can find much stronger requirements within certain fields.

In your example, I believe that you want to make a statement about the probability of success, $p$, in the population based on the estimation from your sample, $\hat{p}$. The procedure for hypothesis testing is then:

  1. State the null hypothesis $p = p_0$, and specify $p_0$, e.g., $p_0 = 0.5$.
  2. Calculate the probability of observing $K \leq np_0$ or the probability of observing $K \geq np_0$ (depending on whether $k < np_0$) from $B(n,p_0)$ with $n=50$ and $p_0$ as stated in $H_0$.
  3. If that probability is lower than your predefined threshold $\alpha$, you have enough evidence to state the opposite of the null hypothesis: $p \neq p_0$.

So, there is no need for Gaussian approximation, you can use the binomial test directly in step 2. Wikipedia has a nice explanation on how to perform a binomial test.

If you for some other reason need a Gaussian approximation, you have to take into account both $n$ and $p$. As you say, your $n$ is not very large, and if your estimated $\hat{p}$ is close to 0 or 1, this makes it even worse. I don't know why you think you can use a Student's $t$ distribution, you simply cannot. The rationale for using a Gaussian approximation is that the Binomial distribution has a limiting Gaussian distribution: when $n\rightarrow \infty$, the Binomial distribution approaches the Gaussian. There is no such relationship between the Binomial distribution and the Student's $t$ distribution.

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    $\begingroup$ In addition, the binomial tails are thinner than the normal tail, so a t distribution cannot be a better approximation! $\endgroup$ – kjetil b halvorsen Feb 24 at 14:29

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