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.

Answer 1

The probability that the diameter falls in the interval from 2499 psi to 2510 psi is 0.00798.

Explanation:

Let's define the random variable,
`X =` "Comprehensive strength of concrete". We have information that
`X` 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,
`X \sim N(2500, 2500)`.

We want to know the probability of the mean of X or
`\bar{X}` that falls in the interval
`[2499;2510]`. From inference theory we know that :

`\bar{X} \sim N(2500, (2500)/(5)) \Rightarrow \bar{X} \sim N(2500,500)`

Now we can find the probability as follows:

`P(2499 \leq \bar{X} \leq 2510) \Rightarrow P((2499 - 2500)/(500) \leq \frac{\bar{X} - 2500}{500} \leq (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 )`

Where
`Z \sim N(0,1)`, then:

`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`