Respuesta :
Answer:
64.56% probability that between 17 and 25 circuits in the sample are defective.
Step-by-step explanation:
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:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]n = 110, p = 0.2[/tex]
So
[tex]\mu = E(X) = np = 110*0.2 = 22[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{110*0.2*0.8} = 4.1952[/tex]
Probability that between 17 and 25 circuits in the sample are defective.
This is the pvalue of Z when X = 25 subtrated by the pvalue of Z when X = 17. So
X = 25
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{25 - 22}{4.1952}[/tex]
[tex]Z = 0.715[/tex]
[tex]Z = 0.715[/tex] has a pvalue of 0.7626.
X = 17
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{17 - 22}{4.1952}[/tex]
[tex]Z = -1.19[/tex]
[tex]Z = -1.19[/tex] has a pvalue of 0.1170.
0.7626 - 0.1170 = 0.6456
64.56% probability that between 17 and 25 circuits in the sample are defective.
