Respuesta :
Answer:
(a) The probability that the box contains at most 20 defective bulbs is 0.9131.
(b) The expected number of defective bulbs in the box is 15.
(c) The standard deviation of the number of defective bulbs is 3.67.
Step-by-step explanation:
Let X = number of defective bulbs.
The probability of selecting a defective bulb is, p = 0.10.
The box consists of n = 150 bulbs.
The random variable X follows a Binomial distribution with parameters n = 150 and p = 0.10.
Normal approximation to binomial can be applied here because:
- np = 150 × 0.10 = 15 > 10
- n(1 - p) = 150 × (1 - 0.10) = 135 > 10
So X follows a Normal distribution with,
Mean: [tex]\mu=np=150\times 0.10=15[/tex]
Standard deviation: [tex]\sqrt{np(1-p)}=\sqrt{150\times 0.10\times (1-0.10)}=3.67[/tex]
(a)
Compute the probability that the box contains at most 20 defective bulbs as follows:
[tex]P(X\leq 20)=P(\frac{X-\mu}{\sigma}\leq \frac{20-15}{3.67}\\=P(Z<1.36)\\=0.9131[/tex]
Thus, the probability that the box contains at most 20 defective bulbs is 0.9131.
(b)
Compute the expected number of defective bulbs in the box as follows:
[tex]E(X)=np\\=150\times 0.10\\=15\\[/tex]
Thus, the expected number of defective bulbs in the box is 15.
(c)
Compute the standard deviation of the number of defective bulbs as follows:
[tex]SD(X)=\sqrt{np(1-p)}=\sqrt{150\times 0.10\times (1-0.10)}=3.67[/tex]
Thus, the standard deviation of the number of defective bulbs is 3.67.
