Mathematics: Simplify: 2/5÷2 3/4 please help no links/files

Simplify: 2/5÷2 3/4 please help no links/files

Ответ:

alaj6878

24.06.2022

Mathematics

a) $P(X \geq 215)$

b) 7.35% probability that at least 215 of the 400 watches are defective.

Step-by-step explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$ , $\sigma = \sqrt{V(X)}$ .

In this problem, we have that:

n = 400, p = 0.5

$\mu = E(X) = np = 400*0.5 = 200$

$\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{400*0.5*0.5} = 10$

(a) Write an expression for the exact probability that at least 215 of the 400 watches are defective.

$P(X \geq 215)$

(b) Approximate the probability, using either the Poisson or normal approximation, whichever is appropriate, that at least 215 of the 400 watches are defective.

Using continuity correction, this is $P(X \geq 215-0.5) = P(X \geq 214.5)$ , which is 1 subtracted by the pvalue of Z when X = 214.5. So