Ball bearings are manufactured with a mean diameter of 5...

Ball bearings are manufactured with a mean diameter of 5 millimeter (mm). Because of variability in the manufacturing process, the diameters of the ball bearings are approximately normally distributed with a standard deviation of $0.02 \mathrm{mm}$ (a) What proportion of ball bearings has a diameter more than $5.03 \mathrm{mm} ?$ (b) Any ball bearings that have a diameter less than 4.95 mm or greater than $5.05 \mathrm{mm}$ are discarded. What proportion of ball bearings will be discarded? (c) Using the results of part (b), if 30,000 ball bearings are manufactured in a day, how many should the plant manager expect to discard? (d) If an order comes in for 50,000 ball bearings, how many bearings should the plant manager manufacture if the order states that all ball bearings must be between $4.97 \mathrm{mm}$ and $5.03 \mathrm{mm} ?$

Ball bearings are manufactured with a mean diameter of 5 millimeter (mm). Because of variability in the manufacturing process, the diameters of the ball bearings are approximately normally distributed with a standard deviation of $0.02 \mathrm{mm}$
(a) What proportion of ball bearings has a diameter more than $5.03 \mathrm{mm} ?$
(b) Any ball bearings that have a diameter less than 4.95 mm or greater than $5.05 \mathrm{mm}$ are discarded. What proportion of ball bearings will be discarded?
(c) Using the results of part (b), if 30,000 ball bearings are manufactured in a day, how many should the plant manager expect to discard?
(d) If an order comes in for 50,000 ball bearings, how many bearings should the plant manager manufacture if the order states that all ball bearings must be between $4.97 \mathrm{mm}$ and $5.03 \mathrm{mm} ?$

Key Concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for a distribution gives the probability that a random variable takes on a value less than or equal to a specified value. For the standard normal distribution, the CDF is used to determine the proportion of observations below (or above) a particular Z-score, which is key for calculating probabilities in quality control and other statistical applications.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, where most of the observations cluster around the central peak and probabilities for values taper off equally on both sides. It is fundamental in statistics because many natural phenomena and measurement errors are approximately normally distributed, making it a powerful tool for inference and decision-making.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion of a set of values. In a normally distributed dataset, about 68% of the values lie within one standard deviation of the mean, and it provides insight into the spread of the data. A smaller standard deviation indicates that the data points tend to be closer to the mean, while a larger one indicates more spread.

Z-score

The Z-score is a numerical measurement that describes a value's position relative to the mean of a group of values, normalized by the standard deviation. It is calculated by subtracting the mean from the value of interest and dividing the result by the standard deviation. This allows different normal distributions to be compared and probabilities to be calculated using the standard normal distribution.

Quality Control in Manufacturing

Quality control in manufacturing involves monitoring and managing the production process to ensure products meet specified standards and tolerances. This includes the use of statistical methods, such as normal distribution analysis, to assess the probability of defects, determine rejection rates, and plan for appropriate production volumes to meet quality requirements.

Transcript

00:01 All right.

00:03 We know that ball bearings have a mean of 5 millimeters and a standard deviation of 0 .02.

00:16 All right.

00:20 In these problems, we're going to be using the normal distribution formula where you take the number you're given, which goes in place of a, subtract the mean, and divide by the standard deviation.

00:33 All right.

00:35 Letter a asks how many ball bearings, what proportion of ball bearings would be greater than 5 .03? all right, so 5 .03 minus 5 over 0 .02.

00:51 All right.

00:53 So this number inside will give us a z value.

00:56 And for this problem, we should get a z value of 1 .5.

01:04 All right.

01:05 Now using a table of z values, i can see that 1 .5 gives me a 93, 1 .5 or less is a 93 .32 % chance of happening.

01:23 All right.

01:23 This is less than, however.

01:27 For this problem, we're asked to find the proportion greater than 5 .03.

01:32 So to do that, we have to subtract 93 .32 from 1.

01:36 So 1 minus 93 .32 is 6 .3 .3 .2 is 6.