Respuesta :
Answer:
The probability that 10 squared centimetres of dust contains more than 10150 particles is 0.067.
Step-by-step explanation:
The Poisson distribution with parameter λ, can be approximated by the Normal distribution, when λ is large say λ > 1,000.
If X follows Poisson (λ) and λ > 1,000 then the distribution of X can be approximated but he Normal distribution.
The mean of the approximated distribution of X is:
μ = λ
The standard deviation of the approximated distribution of X is:
σ = √λ
Thus, if λ > 1,000, then [tex]X\sim N(\mu=\lambda,\ \sigma^{2}=\lambda)[/tex].
Let X = number of asbestos particles in a sample of 1 squared centimetre of dust.
The random variable X follows a Poisson distribution with mean, μ = 1000.
Then the average number of asbestos particles in a sample of 10 squared centimetre of dust will be, [tex]\lambda = 10\times \mu=10\times 1000=10,000[/tex].
Compute the probability that 10 squared centimetres of dust contains more than 10150 particles as follows:
[tex]P(X>10150)=P(\frac{X-\mu}{\sigma}>\frac{10150-10000}{\sqrt{10000}})[/tex]
[tex]=P(Z>1.50)\\=1-P(Z<1.50)\\=1-0.93319\\=0.06681\\\approx0.067[/tex]
*Use a z-table for the probability.
Thus, the probability that 10 squared centimetres of dust contains more than 10150 particles is 0.067.
