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Cloud seeding, a process in which chemicals such as silver iodide and frozen carbon dioxide are introduced by aircraft into clouds to promote rainfall was widely used in the 20 th cen tury. Recent research has questioned its effectiveness [Journal of Atmospheric Research (2010, Vol. 97 (2), pp. 513-525)]. An experiment was performed by randomly assigning 52 clouds to be seeded or not. The amount of rain generated was then measured in acre-feet. Here are the data for the unseeded and seeded clouds: into clouds to promote rainfall was widely used in the 20 th century. Recent research has questioned its effectiveness [Journal of Atmospheric Research (2010, Vol. 97 (2), pp. 513-525)]. An experiment was performed by randomly assigning 52 clouds to be seeded or not. The amount of rain generated was then measured in acre-feet. Here are the data for the unseeded and seeded clouds: Unseeded: \(\begin{array}{lllllllll}81.2 & 26.1 & 95.0 & 41.1 & 28.6 & 21.7 & 11.5 & 68.5 & 345.5 & 321.2\end{array}\) \(\begin{array}{lllllllll}1202.6 & 1.0 & 4.9 & 163.0 & 372.4 & 244.3 & 47.3 & 87.0 & 26.3 & 24.4\end{array}\) \(\begin{array}{llllll}830.1 & 4.9 & 36.6 & 147.8 & 17.3 & 29.0\end{array}\) Seeded: \(\begin{array}{lllllll}274.7 & 302.8 & 242.5 & 255.0 & 17.5 & 115.3 & 31.4 & 703.4 & 334.1\end{array}\) \(\begin{array}{llllllll}1697.8 & 118.3 & 198.6 & 129.6 & 274.7 & 119.0 & 1656.0 & 7.7 & 430.0\end{array}\) \(\begin{array}{llllll}40.6 & 92.4 & 200.7 & 32.7 & 4.1 & 978.0 & 489.1 & 2745.6\end{array}\) Find the sample mean, sample standard deviation, and range of rainfall for (a) All 52 clouds (b) The unseeded clouds (c) The seeded clouds

Step by step solution

01

Organize Data

First, list out all rainfall data for both unseeded and seeded clouds to find the sample mean, standard deviation, and range.

02

Calculate Mean

For each group (all clouds, unseeded, seeded), calculate the mean by summing the rainfall values and dividing by the number of values. For example, the mean of unseeded clouds is calculated as: \[ \text{Mean} = \frac{81.2 + 26.1 + ... + 29.0}{26} \] Repeat similarly for the other groups.

03

Calculate Standard Deviation

For each group, determine the standard deviation using the formula: \[ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \] where \( x_i \) represents each data point, \( \bar{x} \) is the mean of the group, and \( n \) is the number of data points. Perform this calculation for unseeded, seeded, and all clouds.

04

Determine Range

Calculate the range for each group by subtracting the smallest value from the largest value in the dataset. For example, the range of unseeded clouds is: \[ \text{Range} = \text{Maximum} - \text{Minimum} \] Identify the minimum and maximum values in each dataset to compute this.

05

Compile Results

Compile the results from the previous steps to display the mean, standard deviation, and range for all clouds, unseeded clouds, and seeded clouds. This gives a clear view of the comparison.

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Key Concepts

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

Sample Mean

The sample mean is a fundamental concept in statistical analysis used to find the average of a particular dataset. Calculating the mean allows us to understand the central tendency of the values we're observing.
To find the sample mean of a dataset, you sum up all the values and divide the total by the number of data points.
For example, suppose you have a set of numbers representing rainfall amounts for different clouds: \( x_1, x_2, x_3, ..., x_n \). The formula for the mean \( \bar{x} \) is given by:

• \( \bar{x} = \frac{x_1 + x_2 + x_3 + ... + x_n}{n} \)

This calculation gives an accurate centralized value around which all other data points are distributed and helps in understanding the general magnitude of the data.

Sample Standard Deviation

Sample standard deviation provides insight into the spread or variability of a dataset. It indicates how much individual data points deviate from the sample mean. A small standard deviation suggests that the numbers are close to the mean, while a large one indicates more dispersion.
The formula for calculating the sample standard deviation \( \sigma \) is:

• \( \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \)

Here, \( x_i \) refers to each data point, \( \bar{x} \) is the mean of the dataset, and \( n \) represents the number of data points.
Calculating the standard deviation involves three key steps:

• Subtract the mean from each data point and square the result.
• Sum all squared differences.
• Divide by \( n-1 \) and take the square root of the result.

This measure is crucial in understanding how spread out the data is within the experimental design.

Range Calculation

The range of a dataset provides a simple measure of variability by showing the difference between the largest and smallest values. It's an easy way to understand the span of the dataset.
To calculate the range, you simply subtract the smallest value in the dataset from the largest value.
The formula is straightforward:

• \( \text{Range} = \text{Maximum Value} - \text{Minimum Value} \)

This calculation gives a basic sense of the data's extent, capturing the highest point and the lowest point of the data values. However, it's worth noting that the range doesn't provide information about how data points are distributed across the dataset.
It is often used alongside the standard deviation to give a fuller picture of data variability.

Experimental Design

In the study of statistical analysis, experimental design refers to the process of planning an experiment so that valid and objective conclusions can be drawn. It involves structuring trials to ensure observations are valid and unbiased.
Important principles of a good experimental design include:

• Randomization - subjects are randomly assigned to different groups to prevent bias.
• Replication - experiments are repeated to ensure that results are consistent and not due to chance.
• Control - measures are taken to keep conditions the same for all subjects except for the variable being tested.

In the context of the cloud seeding experiment, each cloud was randomly assigned to either the seeded or unseeded group to ensure any differences in rainfall were due to the treatment and not other factors.
This random assignment is a critical step as it minimizes the influence of confounding variables, leading to more reliable and valid results.