Question

Engineers at a computer production plant tested two methods for accuracy in drilling holes into a PC board. They tested how fast they could set the drilling machine by running 10 boards at each of two different speeds. To assess the results, they measured the distance (in inches) from the center of a target on the board to the center of the hole. The data and summary statistics are shown in the table: $$ \begin{array}{lc|l|l|l} & \text { Distance (in.) } & \text { Speed } & \text { Distance (in.) } & \text { Speed } \\ \hline & 0.000101 & \text { Fast } & & 0.000098 & \text { Slow } \\ & 0.000102 & \text { Fast } & & 0.000096 & \text { Slow } \\ & 0.000100 & \text { Fast } & & 0.000097 & \text { Slow } \\ & 0.000102 & \text { Fast } & & 0.000095 & \text { Slow } \\ & 0.000101 & \text { Fast } & & 0.000094 & \text { Slow } \\ & 0.000103 & \text { Fast } & & 0.000098 & \text { Slow } \\ & 0.000104 & \text { Fast } & & 0.000096 & \text { Slow } \\ & 0.000102 & \text { Fast } & & 0.975600 & \text { Slow } \\ & 0.000102 & \text { Fast } & & 0.000097 & \text { Slow } \\ & 0.000100 & \text { Fast } & & 0.000096 & \text { Slow } \\ \hline \text { Mean } & 0.000102 & & \text { Mean } & 0.097647 & \\ \text { StdDev } & 0.000001 & & \text { StdDev } & 0.308481 & \end{array} $$ Write a report summarizing the findings of the experiment. Include appropriate visual and verbal displays of the distributions, and make a recommendation to the engineers if they are most interested in the accuracy of the method.

Short answer

The 'Fast' speed method is more accurate, with a smaller mean distance and less variability.

Step-by-step solution

Understand the Data

The table provides measurements for two sets of holes drilled at different speeds: 'Fast' and 'Slow'. Each speed category has 10 measurements, with descriptive statistics (mean and standard deviation) calculated for each. We will focus on analyzing the accuracy indicated by the mean distance from target.

Analyze Mean Distances

The means of the 'Fast' and 'Slow' speeds are given as 0.000102 and 0.097647 inches, respectively. A smaller mean distance indicates better accuracy, as it means the holes are closer to the target center.

Examine Standard Deviations

The standard deviation indicates variability in measurements. The 'Fast' speed has a standard deviation of 0.000001, while the 'Slow' speed has a much larger standard deviation of 0.308481, suggesting more inconsistency in hole placement with the 'Slow' speed.

Visualize the Distributions

Create box plots or histograms for both 'Fast' and 'Slow' speed distributions. For the 'Fast' speed, the data points would cluster tightly around the mean, while for the 'Slow' speed, the spread would be much wider due to the larger standard deviation.

Make a Recommendation

Based on the analysis, the 'Fast' speed shows both a smaller mean distance and less variability, indicating greater accuracy and consistency in drilling. Therefore, recommend the 'Fast' method if accuracy is the primary concern.

Key concepts

Descriptive Statistics

Descriptive statistics are useful tools to summarize and describe the main features of a data set. In our context, the engineers collected data on how accurately drills placed holes on PC boards at two different speeds, referred to as 'Fast' and 'Slow'. Summarizing this data involves calculating key metrics such as the mean and standard deviation for both sets of data.

The mean, which represents the average value of the measurements, plays a crucial role in understanding accuracy. For the 'Fast' speed, the mean distance from the center target is 0.000102 inches. In contrast, the 'Slow' speed has a mean distance of 0.097647 inches. These numbers suggest that drilling at the 'Fast' speed results in hole placements that are closer to the intended center.

The standard deviation, on the other hand, gives insight into the consistency of the measurements. A lower standard deviation indicates that the data points are closely clustered around the mean, which is a sign of consistency. The 'Fast' speed has a standard deviation of 0.000001, whereas the 'Slow' speed has a much larger standard deviation of 0.308481. This disparity highlights that the 'Fast' speed is not only more accurate but also more consistent.

Distribution Visualization

Visualizations are powerful means to understand and communicate the shape and the spread of data distributions. For this experiment, engineers might use box plots or histograms to represent the distributions of the distance measurements for the 'Fast' and 'Slow' speeds.

In a histogram for the 'Fast' speed, we would expect to see a tall cluster of bars primarily centered around the mean of 0.000102 inches, indicating uniformity and precision in the hole placements. This is consistent with the low standard deviation that signifies little spread in the measurements.

Conversely, a histogram representing the 'Slow' speed would likely display a much wider spread of bars. This spread illustrates greater variability, as indicated by the high standard deviation of 0.308481. Box plots would similarly reveal a tight interquartile range for the 'Fast' speed and a wider one for the 'Slow' speed, showing range and potential outliers more clearly.
These visualizations can quickly convey which method offers more precision and why one might be preferred over the other.

Measurement Accuracy

Measurement accuracy in this experiment is defined as how close the actual hole placements are to the intended target center on the PC board. It is essential for engineers to achieve high accuracy as it relates directly to the quality of the product.

The results of the experiment clearly show that the 'Fast' speed is superior in terms of accuracy. The mean distance from the target center is much smaller for the 'Fast' speed (0.000102 inches) than for the 'Slow' speed (0.097647 inches), demonstrating that the holes drilled at the 'Fast' speed are consistently closer to their targets.

Furthermore, the low standard deviation at the 'Fast' speed further supports this conclusion, as it indicates that the drilling process is not just precise but also reliable, producing consistent results with minimal variation. The recommendation for the engineers would undoubtedly be to use the 'Fast' method if accuracy is a primary goal, due to its impeccable performance in this area.