 # 1)What factors influence the correspondence between the binomial and normal distributions? 1.Twenty percent of individuals who seek psychotherapy will nagasenaz 2021-03-02 Answered
1)What factors influence the correspondence between the binomial and normal distributions? 1.Twenty percent of individuals who seek psychotherapy will recover from their symptoms irrespective of whether they receive treatment. A research finds that a particular type of psychotherapy is successful with 30 out of 100 clients. Using an alpha level of 0.05 as a criterion, what should she conclude about the effectiveness of this psychotherapeutic approach? 2.How does the size of the data set help cut down on the size of the error terms in the approximation process?

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1) Binomial approximation: The normal distribution can be used as an approximation to the binomial distribution, under certain circumstances, namely: If $$\displaystyle{X}\sim{B}{\left({n},{p}\right)}$$ and if n is large and/or p is close to
$$^1/_2$$, then X is approximately N $$(np, npq)$$ (where $$q = 1 - p$$). Continuity Correction: The binomial is a discrete random variables, whereas the normal distribution is continuous. We need to take this into account when we are using the normal distribution to approximate a binomial using a continuity correction. Continuity correction factor table: If $$\displaystyle{P}{\left({X}={n}\right)}\ {u}{s}{e}\ {P}{\left({n}–{0.5}{<}{X}{<}{n}+{0.5}\right)}$$

If $$\displaystyle{P}{\left({X}{>}{n}\right)}\ {u}{s}{e}\ {P}{\left({X}{>}{n}+{0.5}\right)}$$

If $$\displaystyle{P}{\left({X}\leq{n}\right)}\ {u}{s}{e}\ {P}{\left({X}{<}{n}+{0.5}\right)}$$

If $$\displaystyle{P}{\left({X}{<}{n}\right)}\ {u}{s}{e}\ {P}{\left({X}{<}{n}–{0.5}\right)}$$

If $$\displaystyle{P}{\left({X}\geq{n}\right)}\ {u}{s}{e}\ {P}{\left({X}{>}{n}–{0.5}\right)}$$

Continuous normal distribution can be used as an approximation to binomial distribution, the modification is known as continuity correction.