Question

Supppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interest us.

• Suppose n =33 and p=0.20. Can we approximate p^ by a normal distribution?why? Np=
• Nq=

Answer 1

$\text{for a binomial distribution}$

$\text{mean μ of a binomial }$

$\mu=np$

$\text{standard deviation of a binomial}$

$\sigma=\sqrt{np(1-p}$

$\text{For a normal distribution, μ should be 3 standard deviations away from 0 and n}$

$\mu- 3\sqrt{np(1-p)} > 0 \text{ or}$

$np>9(1-p);(1)$

$\mu+3\sqrt{np(1-p)}<n \text{or}$

$n(1-p)>9p;nq>9p(2)$

$np = 33*0.2=6.6$

$np <9(1-0.2)=7.2$

$nq =33*(1-0.2)=26.4$

$nq>9p=1.8$

$\text{сondition(1) not met}$

$\text{it is not possible to approximate } \hat{p}$

$\text{by a normal distribution}$

Answer:it is not possible to approximate a binomial experiment to a normal distribution

np = 6.6 nq =26.4