Question

Refer to the previous exercise. Suppose that there are two machines available for cutting corks. The machine described in the preceding problem produces corks with diameters that are approximately normally distributed with mean \(3 \mathrm{~cm}\) and standard deviation \(0.1 \mathrm{~cm}\). The second machine produces corks with diameters that are approximately normally distributed with mean \(3.05 \mathrm{~cm}\) and standard deviation \(0.01 \mathrm{~cm}\). Which machine would you recommend? (Hint: Which machine would produce fewer defective corks?)

Short answer

The second machine is recommended as it is expected to produce fewer defective corks due to its lower standard deviation, despite its mean being slightly higher.

Step-by-step solution

Understand the Machines' Outputs

First, observe the output of each machine. The first produces corks with a mean diameter of 3 cm and a standard deviation of 0.1 cm, whereas the second machine produces corks with a mean diameter of 3.05 cm and a standard deviation of 0.01 cm. The greater an object's standard deviation, the more spread out it is - this means that corks from the first machine will be more varied in size.

Determine the Bounds for Acceptable Cork Sizes

To determine which machine will produce less defective corks, we first need to introduce a definition for 'defective'. For this exercise, let's say any cork with a diameter under 2.9 cm or over 3.1 cm is considered 'defective'. This gives us a range of allowed values.

Calculate the Proportion of Defective Corks from Each Machine

The next step is to calculate the proportion of defective corks produced by each machine within the given range (2.9 cm - 3.1 cm). For a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. Using these principles, we can calculate the approximate percentage of defective corks from each machine.

Compare the Results and Make a Recommendation

Compare the proportion of defective corks from each machine. The machine producing a lower proportion of defective corks would be the one to recommend. Given the standard deviation and mean for each machine, the second machine should produce fewer defective corks because its results are more clustered around the mean, as suggested by a smaller standard deviation.

Key concepts

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the data points in a distribution deviate, on average, from the mean, or average value. A smaller standard deviation indicates that the values are more closely grouped around the mean, which means there is less variability and the results are more consistent.

For example, in the context of cork production, a machine with a standard deviation of 0.01 cm in the diameter of corks produces results that are very close to the mean diameter. In contrast, a higher standard deviation, like 0.1 cm, suggests a wider range of sizes. This would likely increase the percentage of corks that do not fit within the desired specifications—hence, a higher chance of defective corks.

Understanding standard deviation is essential when comparing two or more sets of data. It helps to make informed decisions, like choosing between two machines based on the consistency of their outputs in manufacturing processes.

Mean

The mean, often referred to as the average, is a measure of the central tendency of a set of numbers. It is calculated by adding up all the values and then dividing by the number of values. In normal distributions, the mean is the point around which the data is symmetrical, and it also represents the peak of the distribution curve.

In practical scenarios like our cork example, the mean diameter of corks produced by each machine is indicative of the overall sizing trend. A mean diameter of 3.05 cm compared to a mean of 3 cm shows us that on average, the corks from the second machine are larger. If the specification states that the ideal cork size is close to 3.05 cm, then the machine with the mean closer to this size will produce more 'ideal' corks, resulting in higher efficiency and lower waste due to fewer defective items.

Defective Corks Analysis

The analysis of defective corks is focused on determining the proportion of corks that do not meet the specified quality standards due to deviations from acceptable size ranges. To do this, we can refer to the properties of the normal distribution curve and the defined acceptable size limits.

In our exercise, corks with a diameter outside the range of 2.9 cm to 3.1 cm are considered defective. Given that normal distribution is symmetrical around the mean, we can use the standard deviation to estimate the percentage of corks falling within this range. The smaller the standard deviation, like in the case of the second machine which is 0.01 cm, the more concentrated the corks will be around the mean, meaning fewer corks will be considered defective.

Therefore, with a smaller standard deviation and a mean diameter that is close to the center of the acceptable range, the second machine is likely to produce a smaller proportion of defective corks. This analysis is crucial in quality control and in reducing material waste in manufacturing processes, leading to cost savings and increased customer satisfaction.