The compressive strength of samples of cement is assumed to be normally distributed with a mean of `6000(kg)/(cm^2)` and a standard deviation of `100(kg)/(cm^2)`.

  1. What is the probability that a sample's strength is less than `5850(kg)/(cm^2)`?
  2. What compressive strength is exceed by 95% of the samples?
  3. What is the probability that at least one sample, among 4 random samples, will have a compressive strength less than `5850(kg)/(cm^2)`?
  4. If the quality control section wants to construct a `bar(x)` chart based on samples of size 6, what will be the `3sigma` control limits for the chart? If the process mean suddenly shifts to `6120(kg)/(cm^2)`, what is the probability that the shift will be detected on the next sample taken? Note that a sample mean falling above the upper control limit or below the lower control limit would signal a shift in the process mean on the `bar(x)` chart.


  1. `P(X<5850)=P(Z<(5850-6000)/100)=P(Z<-1.5)=Phi(-1.5)=0.0668072`.

  2. We need to find such `x` that






  3. Probability that at least one sample will have a compressive strength less than `5850 (kg)/(cm^2)` equals to 1 minus probability that all 4 samples will have a compressive strength greater than `5850(kg)/(cm^2)`.


  4. Since sample is of size 6, then sample mean will be distributed with mean `mu=6000` and standard deviation `sigma=100/sqrt(6)=40.8248`.

    So, `3sigma` control limit is `3*40.8248=122.4745`.

    If sample mean will be outside limits then shift will be detected.

    We need to find