Question

Prediction units. The errors in predicting hurricane tracks (examined in this chapter) were given in nautical miles. An ordinary mile is \(0.86898\) nautical miles. Most people living on the Gulf Coast of the United States would prefer to know the prediction errors in miles rather than nautical miles. Explain why converting the errors to miles would not change the correlation between Prediction Error and Year.

Short answer

Converting errors from nautical miles to miles multiplies by a constant, which doesn't change the correlation.

Step-by-step solution

Convert nautical miles to miles

To convert each prediction error value from nautical miles to ordinary miles, we use the conversion factor: 1 nautical mile = 0.86898 miles. Hence, if the prediction error in nautical miles is denoted by \( x \), then the error in miles is \( y = x \times 0.86898 \).

Understand the correlation property

Correlation measures the strength and direction of a linear relationship between two variables. It is a unitless measure and is not affected by scaling or shifting of units. This means that multiplying all values of one variable by the same positive constant does not change the correlation.

Relate conversion to correlation

When converting from nautical miles to miles, we multiply the prediction errors by 0.86898, a constant. The correlation formula involves standardized values (z-scores), thus multiplying by a constant does not alter their standardized values or the correlation coefficient.

Conclusion

Since the correlation is calculated based on standardized values, and since a constant multiplier does not affect these values, the conversion from nautical miles to miles does not change the correlation between Prediction Error and Year.

Key concepts

Prediction Error

Prediction error refers to the difference between the actual outcome and the predicted result. In the context of hurricane tracks, it represents the discrepancy between where a hurricane was predicted to be and where it actually ended up. This is an important measure because understanding and reducing prediction errors can significantly improve forecasting accuracy.

Sometimes, prediction errors are expressed in specific units, such as nautical miles. In weather forecasting, especially for maritime activities, nautical miles are often preferred. However, this unit may not be as intuitive for people who are more accustomed to ordinary miles. Therefore, converting prediction errors to a more familiar unit can make the results more understandable to a broader audience.

Regardless of the units used, the key takeaway is that the correlation between prediction error and other variables, such as the year, remains unaffected by these conversions. As long as the relationship remains linear, the strength and direction of the correlation stay unaltered.

Conversion Factor

A conversion factor is a ratio that allows you to convert a quantity expressed in one unit to its equivalent in another unit. In the exercise, we encounter a conversion factor for transforming nautical miles into ordinary miles. Specifically, the factor used here is:

• 1 nautical mile = 0.86898 miles

This means for every nautical mile, there are 0.86898 ordinary miles.

To use this conversion factor, you simply multiply the value in nautical miles by 0.86898 to attain the equivalent in ordinary miles. This is a straightforward calculation that aligns variables into a unit that might be more relevant for the audience.

Conversion factors provide a seamless way to bridge the gap between different measurement systems and are handy tools in ensuring numerical data is accessible to multiple audiences, enhancing communication and comprehension.

Units Conversion

Units conversion is the process of converting a measurement from one set of units to another. Understanding how to properly convert units is essential in fields like science and engineering, where precise measurements are critical. In the exercise, converting prediction errors from nautical miles to miles is an example of a units conversion.

Following the concept of unit conversion, when switching from one unit to another, it’s essential to maintain the proportionality of the original measurement. The conversion factor helps ensure that the data retains its original meaning and accuracy.

One vital aspect of unit conversion in relation to correlation is that it doesn't impact the correlation coefficient. This is because correlation is determined by the data's relative positions and standardized values, unaffected by a consistent multiplicative change. Thus, despite changing units, the nature of correlations, such as between prediction error and year, remains unchanged. Understanding this principle can remove confusion and empower more accurate interpretation of data trends.