Question

Are some cities more windy than others? Does Chicago deserve to be nicknamed "The Windy City"? These data are the average wind speeds (in kilometers per hour) for 54 selected cities in the United States \(^{5}\): $$ \begin{array}{rrrrrrrrr} \hline 13.1 & 12.2 & 15.4 & 11.0 & 11.2 & 12.0 & 18.1 & 12.0 & 12.5 \\ 11.2 & 18.4 & 16.8 & 16.5 & 11.8 & 56.2 & 16.0 & 14.9 & 12.6 \\ 13.3 & 16.5 & 15.8 & 11.8 & 12.5 & 11.4 & 14.9 & 12.3 & 16.3 \\ 11.7 & 13.3 & 15.7 & 15.2 & 13.4 & 12.8 & 9.8 & 14.6 & 14.4 \\ 9.9 & 12.6 & 15.2 & 9.8 & 16.3 & 10.6 & 12.6 & 13.4 & 18.4 \\ 15.0 & 15.8 & 7.0 & 10.6 & 15.5 & 15.7 & 12.8 & 17.0 & 13.6 \\ \hline \end{array} $$ a. Construct a relative frequency histogram for the data. (HINT: Choose the class boundaries without including the value \(x=56.2\) in the range of values.) b. The value \(x=56.2\) was recorded at Mt. Washington, New Hampshire. Does the geography of that city explain the observation? c. The average wind speed in Chicago is recorded as 15.8 kilometers per hour. Do you think this is unusually windy?

Short answer

Answer: ____________________

Step-by-step solution

Create a frequency distribution

First, we will create a frequency distribution by grouping the data, except the value x=56.2, into appropriate intervals or classes. Then, calculate the frequency and relative frequency for each class.

Construct a relative frequency histogram

Using the frequency distribution obtained in step 1, we can create a relative frequency histogram. The x-axis will represent the class intervals and the y-axis will represent the relative frequencies. Plot the histogram using the intervals and corresponding relative frequencies.

Discuss the reason for the high wind speed value at Mt. Washington

The value x=56.2, which represents the average wind speed at Mt. Washington, New Hampshire, is significantly higher than the wind speeds for other US cities. It can be explained by its geography. Mt. Washington is a mountain, and its elevation and exposure to stronger winds due to its location might be the reason for higher average wind speeds.

Examine the histogram and compare Chicago's wind speed to other cities

To determine whether Chicago's average wind speed (15.8 kilometers per hour) is unusually windy, examine the relative frequency histogram constructed in step 2. Compare the value to other cities' average wind speeds on the histogram. Examine if Chicago's average wind speed is much higher than the values present in the other classes. If it appears significantly higher compared to other cities, one might say that Chicago deserves to be nicknamed "The Windy City" due to its above-average wind speed.

Key concepts

Relative Frequency Histogram

When trying to understand the distribution of wind speeds across different cities, a relative frequency histogram is an invaluable tool. Imagine a graph that displays rectangles, each representing a different range of wind speeds. These ranges, called class intervals, are plotted along the horizontal axis. The height of each rectangle corresponds to the relative frequency—essentially a measure of how often wind speeds within that interval occur relative to the entire dataset.

To construct this type of histogram, as in our exercise, one would begin by excluding any outlier, such as the unusual value of 56.2 km/h recorded at Mt. Washington. Once the typical data is organized into intervals, each interval's relative frequency is calculated by dividing the number of observations (city wind speeds) in that interval by the total number of observations. These frequencies are then represented as percentages and plotted, giving a clear visual of where the majority of the wind speeds lie—and in our case, determining if Chicago's nickname is justified.

Frequency Distribution

A frequency distribution is the foundation upon which a relative frequency histogram is built. It is a summary that shows the number of occurrences (or frequency) of different data points in a dataset, arranged into class intervals.

For instance, to analyze the average wind speed data, one might create a table with two columns: one for class intervals of wind speeds (e.g., 10-12 km/h, 13-15 km/h, etc.) and one for the frequency of cities that fall within each interval. This method highlights the commonality of different wind speeds, but also simplifies complex data, making trends and patterns easier to spot. With such a table, a frequency distribution, one can quickly ascertain if a data point like Chicago's 15.8 km/h is a common wind speed or if it's on the higher end of wind speeds experienced by cities across the US.

Geographical Effects on Wind Speed

Geography plays a critical role in how wind behaves in a given location. For the outlier in our exercise, Mt. Washington's significantly higher wind speed can be attributed to a number of geographical factors. Factors like altitude, topography, and proximity to bodies of water or mountains can all influence wind speed.

Mt. Washington, being a mountainous area, is more exposed to air movement patterns that result from changes in temperature and pressure at higher elevations. These patterns can accelerate wind speeds, unlike in urban or lowland areas. That unique geographical setting explains why the average wind speed is considerably higher than that of other cities. Comparatively, Chicago's elevated wind speed could be influenced by its location near the Great Lakes, leading to increased airflow. By analyzing geographical factors, one can better understand the variations in wind speed statistics across different regions.