Question

The comprehensive strength of concrete is normally distributed with μ = 2500 psi and σ = 50 psi. Find the probability that a random sample of n = 5 specimens will have a sample mean diameter that falls in the interval from 2499 psi to 2510 psi. Express the final answer to three decimal places (e.g. 0.987).

Answer 1

Answer:

The probability that a random sample of n = 5 specimens will have a sample values that falls in the interval from 2499 psi to 2510 psi = P(2499 < x < 2510) = 0.192

Step-by-step explanation:

For the population,

μ = 2500 psi and σ = 50 psi

But for a sample of n = 5

μₓ = μ = 2500 psi

σₓ = σ/√n = (50/√5)

σₓ = 22.36 psi

So, probability that the value for the sample falls between 2499 psi to 2510 psi

P(2499 < x < 2510)

We normalize/standardize these values firstly,

The standardized score for any value is the value minus the mean then divided by the standard deviation.

For 2499 psi

z = (x - μ)/σ = (2499 - 2500)/22.36 = - 0.045

For 2510 psi

z = (x - μ)/σ = (2510 - 2500)/22.36 = 0.45

To determine the probability the value for the sample falls between 2499 psi to 2510 psi

P(2499 < x < 2510) = P(-0.045 < z < 0.45)

We'll use data from the normal probability table for these probabilities

P(2499 < x < 2510) = P(-0.045 < z < 0.45) = P(z < 0.45) - P(z < -0.045) = 0.674 - 0.482 = 0.192