Question

1. Given a random measurement process (X) with a population mean μ of 12 and population standard deviation σ of 1.5, perform the following calculations while assuming that the underlying random process is Gaussian: a. Give the probability that the next measurement x

i
​
is between 8 and 10 (i.e., P(8 i
​
is greater than 13 (i.e., P(X>12) ) c. What is the precision error of this measurement process?

Answer 1

a. To calculate the probability that the next measurement xi is between 8 and 10, we need to calculate the cumulative probability from the standard normal distribution.

First, we need to standardize the values using the population mean and standard deviation:

Z1 = (8 - μ) / σ
Z2 = (10 - μ) / σ

Substituting the values:
Z1 = (8 - 12) / 1.5 = -2.67
Z2 = (10 - 12) / 1.5 = -1.33

Now, we can use a standard normal distribution table or a calculator to find the cumulative probabilities corresponding to these z-values. The probability that the next measurement xi is between 8 and 10 is given by:

P(8 < xi < 10) = P(Z1 < Z < Z2)

Using the standard normal distribution table or calculator, we find the corresponding probabilities and subtract to get the desired result.

b. To calculate the probability that the next measurement xi is greater than 13, we can follow a similar approach.

Z = (13 - μ) / σ
Z = (13 - 12) / 1.5 = 0.67

P(X > 12) = P(Z > 0.67)

Using the standard normal distribution table or calculator, we can find the probability corresponding to the given z-value.

c. The precision error of the measurement process can be represented by the standard deviation of the population (σ). In this case, the precision error is given as 1.5, which is the population standard deviation. It represents the average amount of variation or spread in the measurements from the population mean.