Respuesta :
Using the normal approximation to the binomial, the probabilities are given by:
a) 0.8485 = 84.85%.
b) 0.0244 = 2.44%.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
- The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with [tex]\mu = np, \sigma = \sqrt{np(1-p)}[/tex].
In this problem, we have that:
- There are 600 bulbs, hence n = 600.
- 5% are defective, hence p = 0.05.
The mean and the standard deviation of the approximation are given, respectively, by:
[tex]\mu = np = 600(0.05) = 30[/tex]
[tex]\sigma = \sqrt{np(1-p)} = \sqrt{600(0.05)(0.95)} = 5.3385[/tex]
Item a:
Using continuity correction, the probability is P(X > 24.5), which is one subtracted by the p-value of Z when X = 24.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{24.5 - 30}{5.3385}[/tex]
[tex]Z = -1.03[/tex]
[tex]Z = -1.03[/tex] has a p-value of 0.1515.
1 - 0.1515 = 0.8485.
0.8485 = 84.85% probability.
Item b:
This probability is P(X < 19.5), which is the p-value of Z when X = 19.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{19.5 - 30}{5.3385}[/tex]
[tex]Z = -1.97[/tex]
[tex]Z = -1.97[/tex] has a p-value of 0.0244.
The probability is of 0.0244 = 2.44%.
More can be learned about the normal distribution at https://brainly.com/question/24663213
