A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 139.1-cm and a standard deviation of 1.1-cm.
Find the probability that the length of a randomly selected steel rod is between 138.3-cm and 142.2-cm.
To find the probability that the length of a randomly selected steel rod is between 138.3 cm and 142.2 cm, we need to calculate the area under the normal distribution curve between these two values.
Step 1: Standardize the values.
We need to convert the given lengths into z-scores by using the formula:
z = (x - μ) / σ
Where:
x = the given value (138.3 cm or 142.2 cm)
μ = the population mean (139.1 cm)
σ = the standard deviation (1.1 cm)
For 138.3 cm:
z1 = (138.3 - 139.1) / 1.1
For 142.2 cm:
z2 = (142.2 - 139.1) / 1.1
Step 2: Calculate the probability.
Once we have the z-scores, we can find the probabilities associated with those z-scores using a standard normal distribution table or a calculator.
Let's assume you are using a standard normal distribution table. Look up the probabilities corresponding to z1 and z2 in the table. The probability associated with z1 will give us the area to the left of 138.3 cm, and the probability associated with z2 will give us the area to the left of 142.2 cm.
Step 3: Calculate the final probability.
To find the probability between 138.3 cm and 142.2 cm, we subtract the probability associated with z1 from the probability associated with z2.
P(138.3 < x < 142.2) = P(z1 < z < z2) = P(z2) - P(z1)
By following these steps and using a standard normal distribution table or a calculator, you can find the probability that the length of a randomly selected steel rod falls between 138.3 cm and 142.2 cm.