Question

A machine operation produces bearings whose diameters are normally distributed, with mean and standard deviation equal to .498 and .002, respectively. If specifications require that the bearing diameter equal .500 inch ±.004 inch, what fraction of the production will be unacceptable?

Short answer

Answer: The fraction of unacceptable production is 0.1600, or 16%.

Step-by-step solution

Identify the parameters for the normal distribution

In this exercise, we have a normal distribution with a mean (μ) of .498 inches and a standard deviation (σ) of .002 inches.

Identify the acceptable range for bearing diameter

The acceptable bearing diameter range is .500 ± .004 inches, which means the bearings must have a diameter between .496 inches and .504 inches to be considered within the specification.

Calculate the z-scores for the acceptable range

We will convert the acceptable range into z-scores using the formula: \[ Z = \frac{X - μ}{σ} \] where X represents the diameter in the acceptable range, μ is the mean, and σ is the standard deviation. We will calculate z-scores for the lower and upper bounds of the acceptable range. For the lower bound (.496 inches): \[ Z_{lower} = \frac{.496- .498}{.002} = -1 \] For the upper bound (.504 inches): \[ Z_{upper} = \frac{.504 - .498}{.002} = 3 \]

Calculate the probability within the acceptable range

We will find the probability using the z-scores that represent the acceptable range. We need to use the standard normal distribution table to find the cumulative probabilities. The cumulative probability for Z = -1 is 0.1587, and for Z = 3, it's 0.9987. Next, we find the probability within the acceptable range by subtracting the two cumulative probabilities: \[ P(Z_{upper}) - P(Z_{lower}) = 0.9987 - 0.1587 = 0.8400 \]

Calculate the fraction of unacceptable production

Now that we have found the probability within the acceptable range, we can calculate the fraction of unacceptable production by subtracting the probability within the acceptable range (0.8400) from 1: \[ 1 - 0.8400 = 0.1600 \] The fraction of unacceptable production is 0.1600, or 16%.

Key concepts

Probability

Probability is a foundational concept in statistics that measures the likelihood of a specific event occurring. Think of it as the chance of something happening when multiple outcomes are possible. For instance, if you toss a fair coin, there is a 50% probability it will land on heads and a 50% probability it will land on tails.

Regarding our bearing production problem, we're interested in the probability that randomly chosen bearings fall outside of the acceptable diameter range. This is what we call the 'fraction of unacceptable production'. We calculate probability by looking at the total number of possible outcomes and determining how many of those outcomes fulfill our criteria. For continuous data, like bearing diameters that are normally distributed, we use the properties of the normal distribution to find these probabilities.

Z-scores

Z-scores, or standard scores, are a way to describe a data point's position within a distribution. A z-score tells us how many standard deviations an element is from the mean. It's a method of making different sets of data comparable by standardizing them.

In the exercise, we calculated z-scores for the lower and upper bounds of the acceptable bearing diameter, which are essentially the number of standard deviations those bounds are from the mean diameter. A z-score of -1 for the lower bound means it's one standard deviation below the mean, and a z-score of 3 for the upper bound means it's three standard deviations above the mean. These scores allow us to use the standard normal distribution table, which then helps us find the probability associated with each score.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

In our case, the standard deviation of the bearing diameters is .002 inches, suggesting that the diameters are fairly consistent, deviating by only .002 inches on average from the mean diameter of .498 inches. When we calculate z-scores, we're actually seeing how many 'standard deviations' the specifications are from our process mean, which is critical to determining the likelihood (probability) of a bearing being out of spec.

Standard Normal Distribution Table

The standard normal distribution table, often referred to as the z-table, is a reference that shows the cumulative probability associated with each z-score. It represents the normal distribution, which is a bell-shaped curve symmetrical around the mean, with a mean of zero and a standard deviation of one.

To use the table, we look up our calculated z-score and find the corresponding probability. This probability tells us the likelihood that a randomly selected value from the distribution is less than our given z-score. For instance, in the exercise for the z-score of -1, we found a cumulative probability of 0.1587, and for a z-score of 3, we found 0.9987. By finding these cumulative probabilities, we can determine what fraction of our bearings will likely fall within the acceptable range and, inversely, the fraction expected to be unacceptable.