Question

The compressive strengths of 40 concrete cylinders have an estimated mean of 60.14 MPa and standard deviation of 5.02 MPa. Assuming the strength to be Normally distributed, what is the probability that 10 cylinders will each have strengths between 45 and 75 MPa?

Answer 1

Answer: 0.9972

Explanation:

Let x be a random variable that represents the strength of concrete cylinders.

Given: The compressive strengths of 40 concrete cylinders have an estimated mean
`(\mu)` of 60.14 MPa and standard deviation
`(\sigma)` of 5.02 MPa.

The probability that 10 cylinders will each have strengths between 45 and 75 MPa :

`P(45<\overline{x}<75)=P((45-60.14)/(5.02)<\frac{\overline{x}-\mu}{\sigma}<(75-60.14)/(5.02))\\\=P(-3.016<z<2.96)\\\\=P(z<2.96)-P(z<-3.016)\\\\=P(z<2.96)-(1-P(z<3.016))\\\\=0.9985-(1-0.9987)\ \ \ [\text{by p-value table}]\\\\= 0.9972`

Hence, The required probability = 0.9972