5 votes
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Hypothesis test, finding p-value

I think the Central Limit Theorem is a poor approximation here for 5 samples. The exact probability is $P(T \ge t) = P(5T \ge 4) = P(5T = 5) + P(5T = 4) = \frac{1}{2^5} (\binom{5}{0} + \binom{5}{1}) = ...
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  • 81.8k
2 votes
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2500 people have a car insurance there are 1500 women and 1000 men. Estimate probability that in the next year there will be at least 25 car accidents

There are versions of CLT that does not require identically distributed random variables. See, e.g, wikipedia and http://personal.psu.edu/drh20/asymp/fall2002/lectures/ln04.pdf. Let $W_i$, $i = 1,\...
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  • 4,244
2 votes
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Strong invariance principles

The sequence $\{S_n\}$ has the same almost sure properties as the sequence $\{S_n'\}$ and more generally for any measurable set $A$ in sequence space the chance that $\{S_n\}$ is in $A$ equals the ...
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1 vote

Hypothesis test, finding p-value

$ H_{0}: X \sim \mathcal{N}( 0,5;\ \ 5\cdot 0,5 \cdot (1-0,5))= \mathcal{N}(0,5;\ \ 1,25).$ $ p-value = P(\{X >0,8\}) = 1 - P(\{Z\leq \frac{0,8-0,5}{\sqrt{1,25}}\}) = 1 -\phi(0,2683)= 0,3942.$
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  • 19
1 vote
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Confused on approximation of Binomial using Normal distribution

Here I have plotted the PMF of the binomial distribution with parameters $n = 100$, $p = 0.12$, which is represented as the blue shaded region, and the normal approximation with mean $\mu = np = 12$ ...
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