Answer:
[tex](a)\ P(x \ge 215)[/tex]
[tex](b)\ P(x \ge 214.5) = 0.07353[/tex]
Step-by-step explanation:
Given
[tex]p = 0.50[/tex] ---- proportion of watches with defects
[tex]n = 400[/tex] --- Number of watches
Solving (a): Represent at least 215 of 400 are defective
In inequalities, at least means: [tex]\ge[/tex]
So, the probability is represented as: [tex]P(x \ge 215)[/tex]
Solving (b): Calculate [tex]P(x \ge 215)[/tex]
Normal or Poisson: Normal distribution is characterized by 2 parameters [tex]\mu[/tex] and [tex]\sigma[/tex].
These two parameters can be easily calculated from the given parameters in the question. So, we solve using normal distribution
Start by calculating the mean
[tex]\mu =np[/tex]
[tex]\mu = 0.50 * 400[/tex]
[tex]\mu = 200[/tex]
Calculate standard deviation
[tex]\sigma = \sqrt{\mu (1 - p)[/tex]
[tex]\sigma = \sqrt{200 * (1 - 0.50)[/tex]
[tex]\sigma = \sqrt{200 * 0.50[/tex]
[tex]\sigma = \sqrt{100[/tex]
[tex]\sigma = 10[/tex]
By continuity correction, we have:
[tex]x \to x - 0.5[/tex]
[tex]x \to 215 - 0.5[/tex]
[tex]x \to 214.5[/tex]
So, we have:
[tex]P(x \ge 215) = P(x \ge 214.5)[/tex]
Calculating [tex]P(x \ge 214.5)[/tex], we have:
[tex]P(x \ge 214.5) = 1 - P(x < z)[/tex]
Calculate z score
[tex]z = \frac{x - \mu}{\sigma}[/tex]
[tex]z = \frac{214.5 - 200}{10}[/tex]
[tex]z = \frac{14.5}{10}[/tex]
[tex]z = 1.45[/tex]
So, we have:
[tex]P(x \ge 214.5) = 1 - P(x < 1.45)[/tex]
Using the z score probability table, we have:
[tex]P(x < 1.45) = 0.92647[/tex]
So, we have:
[tex]P(x \ge 214.5) = 1 - 0.92647[/tex]
[tex]P(x \ge 214.5) = 0.07353[/tex]
