Np>5 and nq>5, estimate P(at least 11) with n=13 and p = 0.6 by using the normal distributions as an approximation to the binomial distribution. if np<5 or nq<5 then state the normal approximation is not suitable. p (at least 11) = ?

Question
Normal distributions
asked 2021-01-27
Np>5 and nq>5, estimate P(at least 11) with n=13 and p = 0.6 by using the normal distributions as an approximation to the binomial distribution. if \(\displaystyle{n}{p}{<}{5}{\quad\text{or}\quad}{n}{q}{<}{5}\)</span> then state the normal approximation is not suitable.
p (at least 11) = ?

Answers (1)

2021-01-28
Step 1
Given n=12 and p=0.6.
Then,
Mean = np
\(\displaystyle={13}\times{0.6}\)
=7.8
Standard deviation \(\displaystyle=\sqrt{{{n}{p}{\left({1}-{p}\right)}}}\)
\(\displaystyle=\sqrt{{{7.8}{\left({1}-{0.6}\right)}}}\)
\(\displaystyle=\sqrt{{{3.12}}}\)
=1.7663
Step 2
So,
\(\displaystyle{P}{\left({X}\ge{11}\right)}={P}{\left(\frac{{{X}-{m}{e}{a}{n}}}{{{s}{\tan{{d}}}{a}{r}{d}{d}{e}{v}{i}{a}{t}{i}{o}{n}}}\ge\frac{{{11}-{m}{e}{a}{n}}}{{{s}{\tan{{d}}}{a}{r}{d}{d}{e}{v}{i}{a}{t}{i}{o}{n}}}\right)}\)
\(\displaystyle={P}{\left({Z}\ge\frac{{{11}-{7.8}}}{{{1.7663}}}\right)}\)
\(\displaystyle={P}{\left({Z}\ge{1.81}\right)}\) From the right tailed z - table.
=0.4649
Therefore, p (at least 11)=0.4649
0

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