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.

Answer:The probability that the diameter falls in the interval from 2499 psi to 2510 psi is 0.00798.Step-by-step explanation:Let's define the random variable, [tex]X =[/tex] "Comprehensive strength of concrete". We have information that [tex]X[/tex] is normally distributed with a mean of 2500 psi and a standard deviation of  50 psi (or a variance of 2500 psi). In other words, [tex]X \sim N(2500, 2500)[/tex].We want to know the probability of the mean of X or [tex]\bar{X}[/tex] that falls in the interval [tex][2499;2510][/tex]. From inference theory we know that :[tex]\bar{X} \sim N(2500, \frac{2500}{5}) \Rightarrow \bar{X} \sim N(2500,500)[/tex]Now we can find the probability as follows:[tex]P(2499 \leq \bar{X} \leq 2510) \Rightarrow P(\frac{2499 - 2500}{500} \leq \frac{\bar{X} - 2500}{500} \leq \frac{2499 - 2500}{500} ) \Rightarrow\\\Rightarrow P(-0.002 \leq \frac{\bar{X} - 2500}{500} \leq 0.02 ) \Rightarrow P(-0.002 \leq Z \leq 0.02 )[/tex]Where [tex]Z \sim N(0,1)[/tex], then:[tex]P(-0.002 \leq Z \leq 0.02 ) \approx P(0 \leq Z \leq 0.02 ) = P(Z \leq 0.02 ) - P(Z \leq 0) \\P(0 \leq Z \leq 0.02 ) = 0.50798 - 0.5 = 0.00798[/tex]