Respuesta :
Answer:
0.1230 = 12.30% probability that your stapler will jam less than 23 times
Step-by-step explanation:
I am going to use the normal approximation to the binomial distribution 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:
[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]p = 0.109, n = 260[/tex]
So
[tex]\mu = E(X) = np = 260*0.109 = 28.34[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{260*0.109*0.891} = 5.025[/tex]
Out of 260 papers stapled, what is the probability that your stapler will jam less than 23 times?
This is P(X < 23).
Using continuity correction, this is P(X < 23-0.5) = P(X < 22.5), which is the pvalue of Z when X = 22.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{22.5 - 28.34}{5.025}[/tex]
[tex]Z = -1.16[/tex]
[tex]Z = -1.16[/tex] has a pvalue of 0.1230
0.1230 = 12.30% probability that your stapler will jam less than 23 times
The probability that your stapler will jam less than 23 times is 12.30 %
Probability :
It is given that, On any given attempt to staple, it seems to independently jam 10.9% of the time.
Probability of success[tex]p=0.109,n=260[/tex]
Mean [tex]\mu=np=260*0.109=28.34[/tex]
standard deviation [tex]\sigma=\sqrt{np(1-p)} =5.025[/tex]
We have to find the probability that your stapler will jam less than 23 times.
It means that, [tex]P(X < 23)[/tex], this is the p value of z when [tex]X=23-0.5=22.5[/tex]
To find z- value,
[tex]z=\frac{X-\mu}{\sigma} \\\\z=\frac{22.5-28.34}{5.025} =-1.16[/tex]
From z value table, for [tex]z=-1.16[/tex], p- value is [tex]0.1230[/tex].
Thus, the probability that your stapler will jam less than 23 times is [tex]12.30\%[/tex]
Learn more about the standard deviation here:
https://brainly.com/question/12402189
