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I'm working on a stats project, but in the assignment there is some notation I haven't been able to find in the book or googling. in the below, I don't know which equation N(u,o^2) refers to; part1 with mean $= E(X)$ and $\sigma^2 = \text{VAR}(X)$ for $B \sim \text{Bernoulli}(p)$ with $p = 0.4$.

I'm attempting to find out the probability of a mean being less than the sample proportion (in this case, 0.4) of a binomial distribution.

I already have a simulation approximation, an exact value, but now I also need one via the CLT... but I'm a tad lost.

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    $\begingroup$ stats.stackexchange.com/… $\endgroup$
    – whuber ♦
    Commented Apr 2, 2017 at 18:07
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    $\begingroup$ The sample mean is not magically normal when the sample size exceeds 30 as the insert suggests. Where did you get that the observations were Bernoulli? I should note that there are population distributions for which the central limit theorem doesn't even apply (e.g. the Cauchy distribution). $\endgroup$
    – Michael R. Chernick
    Commented Apr 2, 2017 at 18:14
  • $\begingroup$ Ah, from the rest of the project description. Didn't want to upload the entire thing. This was from part 6. Already simulated said binomial distribution 10,000 times, and this was to compare to that. $\endgroup$
    – Alanek
    Commented Apr 2, 2017 at 21:44

2 Answers 2

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$N(μ,σ^2)$ means a normal distribution with mean $μ$ and variance $σ^2$.

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I don't know which equation $N(\mu,\sigma^2)$ refers to;

$\hat{p}_{50}$ is a random variable. It is not an equation. There are a few ways to "deal" with random variables. Here are some:

  1. It's probability density function or probability mass function
  2. It's cumulative distribution function (you probably want this one for this example)
  3. It's moment generating function
  4. It's label ($\hat{p}_{50}$ is a label in your case)

In your case you want $P(\hat{p}_{50} > p) = 1 - F_{\hat{p}_{50}}(p)$. When I write $F_{\hat{p}_{50}}(p)$ I am referring to the cumulative distribution function of your random variable $\hat{p}_{50}$. The central limit theorem lets you approximate it's CDF with a normal CDF. This is handy because a lot of computers have the capability to evaluate the normal cdfs with different means and different variances/standard deviations.

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