Respuesta :
Answer: 0.6827
Step-by-step explanation:
Given : The resistance of a strain gauge is normally distributed .
Population mean : [tex]\mu=100[/tex]
Standard deviation : [tex]\sigma=0.3[/tex]
Let x be the random variable that represents the resistance of a strain gauge is normally distributed .
z-score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]
To meet the specification, the resistance must be within the range 100±0.7 ohms i.e. (99.3,100.7).
For x = 100.7
[tex]z=\dfrac{100.7-100}{0.7}=1[/tex]
For x = 99.3
[tex]z=\dfrac{99.3-100}{0.7}=-1[/tex]
By using the standard normal distribution table, The proportion of gauges is acceptable is given by:-
[tex]P(99.3<x<100.7)=P(-1<z<1)=P(z<1)-P(z<-1)\\\\=0.8413447-0.1586553=0.6826894\approx0.6827[/tex]
Hence, the proportion of gauges is acceptable=0.6827
The concept of a normal distribution is applied, then the proportion of the gauge is acceptable is 0.6827.
What is a normal distribution?
It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called the bell curve.
Given
Mean ([tex]\mu[/tex]) is 100
The standard deviation ([tex]\sigma[/tex]) is 0.3
Let x be the random variable that represents the resistance of the strain gauge is normally distributed.
Z-score is given by
[tex]\rm z = \dfrac{x - \mu}{\sigma}[/tex]
The resistance must be in a range [tex]\rm 100 \pm 0.7 \ ohms[/tex] that is (99.3, 100.7)
For x = 99.3, then z-score will be
[tex]\rm z = \dfrac{99.3 - 100}{0.7}\\\\z = -\dfrac{0.7}{0.7}\\\\z = -1[/tex]
For x = 100.7, then z-score will be
[tex]\rm z = \dfrac{100.7- 100}{0.7}\\\\z = \dfrac{0.7}{0.7}\\\\z = 1[/tex]
By using the standard normal distribution table. The proportion of gauges is acceptable given by
[tex]\rm P(99.3
Thus, the proportion of the gauge is acceptable is 0.6827.
More about the normal distribution link is given below.
https://brainly.com/question/12421652
