Question

The defect length of a corrosion defect in a pressurized steel pipe is normally distributed with mean value 33 mm and standard deviation 7.9 mm. (a) What is the probability that defect length is at most 20 mm

Answer 1

The probability that defect length is at most 20 mm is 0.0495.

Explanation:

We are given that the defect length of a corrosion defect in a pressurized steel pipe is normally distributed with a mean value of 33 mm and a standard deviation of 7.9 mm.

Let X = the defect length of a corrosion defect in a pressurized steel pipe

The z-score probability distribution for the normal distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = mean defect length = 33 mm

`\sigma` = standard deviation = 7.9 mm

So, X ~ Normal(
`\mu=33 \text{ mm}, \sigma^(2) = 7.9^(2) \text{ mm}`)

Now, the probability that defect length is at most 20 mm is given by = P(X
`\leq` 20 mm)

P(X
`\leq` 20 mm) = P(
`(X-\mu)/(\sigma)`
`\leq`
`(20-33)/(7.9)` ) = P(Z
`\leq` -1.65) = 1 - P(Z < 1.65)

= 1 - 0.9505 = 0.0495

The above probability is calculated by looking at the value of x = 1.65 in the z table which has an area of 0.9505.

Answer 2

0.49926

Explanation:

We solve for this using z score formula

z-score is

z = (x-μ)/σ,

where x is the raw score

μ is the population mean

σ is the population standard deviation

The probability that defect length is at most 20 mm is calculated as:

x = 20mm, μ = 33 mm, σ = 7.9mm

z = 20 - 33/7.9

= -13/7.9

= -1.64557

Obtaining the Probability value from Z-Table:

Probability (At most 20mm) = P(x ≤ 20mm) P(z = -1.64557)

P(x ≤ 20) = 0.049926

Therefore, the probability that defect length is at most 20 mm is 0.049926