Respuesta :
Answer:
The probability that fewer than 10 in a sample of 200 pills will be ineffective is 0.4364
∴ P(X<10)=0.4364
Step-by-step explanation:
Given that the total number of sample pills is 200
ie., n=200
To find the probability that fewer than 10 in a sample of 200 pills will be ineffective:
Let us assume it success if a pill is ineffective.
The probability of success in each trial is [tex]p=5\%[/tex]
[tex]p=5\%[/tex]
[tex]p=\frac{5}{100}[/tex]
[tex]=0.05[/tex]
∴ p=0.05
We know that the total probability is p+q=1
The probability of failure is q
q=1-p
q=1-0.05
∴ q=0.95
Let X be the random variable of the number of ineffective pills in a sample of 200 pills.
Hence X has Binomial distribution with parameter n=200 and p=0.05
The formula for Mean in Binomial distribution is
[tex]\mu=np[/tex]
Substitute the values in the above formula we get
[tex]\mu=200(0.05)[/tex]
∴ [tex]\mu=10[/tex]
The formula for Standard deviation in Binomial distribution is
[tex]\sigma=\sqrt{npq}[/tex]
Substitute the values in the above formula we get
[tex]\sigma=\sqrt{(200)(0.05)(0.95)}[/tex]
[tex]=\sqrt{9.5}[/tex]
[tex]=3.082[/tex]
∴ [tex]\sigma=3.082[/tex]
Now we have to find the probability that fewer than 10 in a sample of 200 pills will be ineffective.
That is to find the area to the left of x=9.5
The formula is [tex]z=\frac{x-\mu}{\sigma}[/tex]
Substitute the values in the formula we get
[tex]z=\frac{9.5-10}{3.082}[/tex]
[tex]=\frac{-0.5}{3.082}[/tex]
[tex]=-0.16[/tex]
∴ [tex]z=-0.16[/tex]
Now P(X<10)=P(Z<-0.16)
=0.4364
