A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 122.7-cm and a standard deviation of 0.5-cm. Find the probability that the length of a randomly selected steel rod is between 122.8-cm and 123.2-cm.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Steel Rod Length Probability Calculation**
A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 122.7 cm and a standard deviation of 0.5 cm.
**Problem Statement:**
Find the probability that the length of a randomly selected steel rod is between 122.8 cm and 123.2 cm.
**Solution:**
To find this probability, we will use the properties of the normal distribution and standard scores (z-scores). A z-score is calculated as follows:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \( X \) is the value,
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation.
Calculate the z-scores for 122.8 cm and 123.2 cm, and use the standard normal distribution table to find the probability.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24a3da1e-8078-4e69-95e8-f18f95f918aa%2Fad97de14-0f68-4f62-ae1c-7c8efab2033c%2Fh7b80dl_processed.jpeg&w=3840&q=75)
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