91Ó°ÊÓ

The Grand Canyon and the Colorado River are beautiful, rugged, and sometimes dangerous. Thomas Myers is a physician at the park clinic in Grand Canyon Village. Dr. Myers has recorded (for a 5 -year period) the number of visitor injuries at different landing points for commercial boat trips down the Colorado River in both the Upper and Lower Grand Canyon (Source: Fateful Journey by Myers, Becker, Stevens). Upper Canyon: Number of Injuries per Landing Point Between North Canyon and Phantom Ranch \(\begin{array}{lllllllllll}2 & 3 & 1 & 1 & 3 & 4 & 6 & 9 & 3 & 1 & 3\end{array}\) Lower Canyon: Number of Injuries per Landing Point Between Bright Angel and Lava Falls \(\begin{array}{llllllllllllll}8 & 1 & 1 & 0 & 6 & 7 & 2 & 14 & 3 & 0 & 1 & 13 & 2 & 1\end{array}\) (a) Compute the mean, median, and mode for injuries per landing point in the Upper Canyon. (b) Compute the mean, median, and mode for injuries per landing point in the Lower Canyon. (c) Compare the results of parts (a) and (b). (d) The Lower Canyon stretch had some extreme data values. Compute a \(5 \%\) trimmed mean for this region, and compare this result to the mean for the Upper Canyon computed in part (a).

Step by step solution

01

Calculate Mean for Upper Canyon

To find the mean, add all the numbers together and divide by the total number of data points. The data points are: 2, 3, 1, 1, 3, 4, 6, 9, 3, 1, 3.Sum: \( 2 + 3 + 1 + 1 + 3 + 4 + 6 + 9 + 3 + 1 + 3 = 36 \)Number of data points: 11Mean: \( \frac{36}{11} \approx 3.27 \) injuries per landing point.

02

Calculate Median for Upper Canyon

To find the median, first sort the data in ascending order: 1, 1, 1, 2, 3, 3, 3, 3, 4, 6, 9. Since there are 11 data points, the median is the 6th value: 3 injuries.

03

Calculate Mode for Upper Canyon

The mode is the number that appears most frequently in the dataset. In the Upper Canyon data, the number 3 appears four times, more than any other number. Thus, the mode is 3 injuries.

04

Calculate Mean for Lower Canyon

The data points for the Lower Canyon are: 8, 1, 1, 0, 6, 7, 2, 14, 3, 0, 1, 13, 2, 1.Sum: \( 8 + 1 + 1 + 0 + 6 + 7 + 2 + 14 + 3 + 0 + 1 + 13 + 2 + 1 = 59 \)Number of data points: 14Mean: \( \frac{59}{14} \approx 4.21 \) injuries per landing point.

05

Calculate Median for Lower Canyon

First, sort the Lower Canyon data: 0, 0, 1, 1, 1, 1, 2, 2, 3, 6, 7, 8, 13, 14.With 14 data points, the median is the average of the 7th and 8th values: \( \frac{2+2}{2} = 2 \) injuries.

06

Calculate Mode for Lower Canyon

The mode is the number that appears most frequently. In the Lower Canyon data, the number 1 appears four times, more than any other number. Therefore, the mode is 1 injury.

07

Compare Upper and Lower Canyon Statistics

For the Upper Canyon: Mean = 3.27, Median = 3, Mode = 3. For the Lower Canyon: Mean = 4.21, Median = 2, Mode = 1. The Lower Canyon has a higher mean, reflecting greater average injuries per point, but a lower median and mode than the Upper Canyon.

08

Compute 5% Trimmed Mean for Lower Canyon

First, remove the top 5% and bottom 5% of the data. For 14 data points, remove 0.7 points from each end, which rounds to removing 1 point from each end.Sorted data without extremes: 0, 1, 1, 1, 2, 2, 3, 6, 7, 8, 13.Sum of new data: \( 0 + 1 + 1 + 1 + 2 + 2 + 3 + 6 + 7 + 8 + 13 = 44 \)Number of data points: 12Trimmed mean: \( \frac{44}{12} \approx 3.67 \), which is closer to the Upper Canyon's mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean

The mean is a measure of central tendency, commonly known as the "average." It is calculated by adding up all the observations and then dividing by the total number of observations.
For instance, in the case of the Upper Canyon injuries, we take the sum of all the number of injuries recorded at each landing point, which totals to 36, and divide it by 11 (the total number of landing points), resulting in a mean of approximately 3.27 injuries per landing point.
In practical terms, the mean gives us a quick overview of the average number of injuries that occur at each landing point. It's a useful metric when you want a single number to represent a dataset, but it doesn't show how the injuries are distributed across the canyon.

Median

The median is the middle value in a dataset when the numbers are arranged in order. It is a measure of central tendency that is robust to outliers.
In the Upper Canyon example, we first arrange the injuries numbers in ascending order: 1, 1, 1, 2, 3, 3, 3, 3, 4, 6, 9. With 11 data points, the 6th number in this ordered list is 3, making it the median.
The median is particularly handy when dealing with skewed data because it is not affected by extreme values. For instance, even if one landing point had a significantly higher or lower number of injuries, the median would remain the same, providing a more reliable indicator of central tendency in skewed distributions.

Mode

The mode is the most frequently occurring number in a dataset. It's useful for identifying which number is most common.
In the Upper Canyon dataset, the number 3 appears four times, more than any other number. Hence, the mode for this dataset is 3.
Knowing the mode can be helpful to find the most typical case within a dataset. It is especially relevant in datasets where you need to see how often a specific number appears, like identifying the most common number of injuries per landing point.

Trimmed Mean

The trimmed mean is a version of the mean that excludes outliers. By removing a small percentage of the largest and smallest values, you can calculate a more balanced mean.
In the Lower Canyon example, with 14 data points, a 5% trimmed mean involves removing approximately one data point from both ends of the sorted list: the smallest and the largest. This results in excluding extreme values that could skew the average. The recalculated sum is 44, with 12 data points left, giving a trimmed mean of approximately 3.67.
The trimmed mean is valuable when the dataset includes outliers that significantly impact the mean. It gives a clearer picture of the typical value without letting extreme values distort the overall average.