Respuesta :
Answer:
0.159 is the probability that the total length after insertion is between 34.5 inch and 35 inch.
Step-by-step explanation:
We are given the following information in the question:
The length of the first piece is normally distributed with mean value 20 inch and standard deviation 0.5 inch.
The length of the second piece is a normal with mean and standard deviation 15 inch and 0.4 inch respectively.
The amount of overlap is normally distributed with mean value 1 inch and standard deviation 0.1 inch.
The lengths and amount of overlap are independent of one another.
Let the total length of PVC pipe be represented by Y.
The sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances
Then, Y follows a normal distribution with
Y = Length of first pipe + Length of second Pipe - Overlap
[tex]\text{Mean} = \text{Mean of first pipe} + \text{Mean of second pipe} - \text{Mean of overlap}\\= 20 + 15 - 1 = 34[/tex]
[tex]\text{Variation} = \text{Variation of first pipe} + \text{Variation of second pipe} - \text{Variation of overlap}\\= (0.5)^2 + (0.4)^2 + (0.1)^2 = 0.42[/tex]
[tex]\text{Standard deviation} = \sqrt{Variance} = \sqrt{0.42} = 0.648[/tex]
P(total length after insertion is between 34.5 inch and 35 inch)
[tex]P(34.5 \leq y \leq 35) = P(\displaystyle\frac{34.5 - 34}{0.648} \leq z \leq \displaystyle\frac{35-34}{0.648}) = P(0.7716 \leq z \leq 1.5432)\\\\= P(z \leq 1.5432) - P(z < 0.7716)\\= 0.939 -0.780 = 0.159 = 15.9\%[/tex]
[tex]P(34.5 \leq y \leq 35) = 15.9\%[/tex]
0.159 is the probability that the total length after insertion is between 34.5 inch and 35 inch.
