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Chapter 3: Problem 19

The Los Angeles Times regularly reports the air quality index for various areas of Southern California. A sample of air quality index values for Pomona provided the following data: \(28,42,58,48,45,55,60,49,\) and 50 a. Compute the range and interquartile range. b. Compute the sample variance and sample standard deviation. c. \(\quad\) A sample of air quality index readings for Anaheim provided a sample mean of 48.5 a sample variance of \(136,\) and a sample standard deviation of \(11.66 .\) What comparisons can you make between the air quality in Pomona and that in Anaheim on the basis of these descriptive statistics?

Short Answer

Expert verified
Pomona has more consistent air quality than Anaheim, but similar averages.

Step by step solution

01

Calculate the Range

To find the range, subtract the smallest value in the data set from the largest value. Here the smallest value is 28 and the largest is 60. \[ \text{Range} = 60 - 28 = 32 \]
02

Calculate the Interquartile Range (IQR)

First, order the data: 28, 42, 45, 48, 49, 50, 55, 58, 60. Find the first quartile \(Q_1\), which is the median of the first half of the data (42, 45, 48, 49). \(Q_1 = 45\). The third quartile \(Q_3\) is the median of the second half (50, 55, 58, 60), thus \(Q_3 = 58\).\[ \text{IQR} = Q_3 - Q_1 = 58 - 45 = 13 \]
03

Calculate the Mean

The mean is the sum of all the data points divided by the number of data points. \[ \text{Mean} = \frac{28 + 42 + 58 + 48 + 45 + 55 + 60 + 49 + 50}{9} = 48.33 \]
04

Calculate the Sample Variance

Calculate each deviation from the mean, square it, then divide by \(n-1\): \[ s^2 = \frac{(28 - 48.33)^2 + (42 - 48.33)^2 + \, ... + (50 - 48.33)^2}{8} = 100.42 \]
05

Calculate the Sample Standard Deviation

The standard deviation is the square root of the variance: \[ s = \sqrt{100.42} = 10.02 \]
06

Compare Pomona and Anaheim

Pomona and Anaheim have similar means (48.33 and 48.5, respectively), but Pomona has a smaller variance (100.42 versus 136) and standard deviation (10.02 versus 11.66), indicating more consistent air quality.

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Key Concepts

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

Air Quality Index
The Air Quality Index (AQI) is a crucial measurement for understanding how clean or polluted the air is in a given location. It presents a clear scale of air quality from 0 to 500, where lower numbers indicate better air quality, and higher numbers suggest more pollution and associated health risks. Monitoring air quality through AQI measurements helps people make informed decisions about outdoor activities, especially for those with respiratory issues. Pomona's air quality index data ranged broadly, but it allows residents to ascertain average air conditions, which could guide them in daily activities and long-term health considerations. Understanding changes in air quality can also help highlight wider environmental trends, perhaps eliciting community concern and actions to improve air standards.
Range and Interquartile Range
The range is one of the simplest methods to measure data spread; it is simply the difference between the maximum and minimum values. Calculating the range for air quality index readings in Pomona, we find a value of 32 ( 60 - 28), which provides a quick snapshot of the variability. On the other hand, the interquartile range (IQR) offers a more robust insight into the data by focusing on the middle 50% of values. For Pomona's air quality, once the data was set in ascending order, the first quartile ( Q_1=45) and third quartile ( Q_3=58) were determined. The IQR is thus 13 ( 58 - 45). This value tells us how spread out the middle half of the data is, girding against any erratic outliers that might distort our perceptions of ordinary air quality.
Sample Variance and Standard Deviation
In statistical analysis, variance and standard deviation are indispensable tools for understanding data distribution. The variance measures how far each number in a set is from the mean, providing an aggregated gauge of data spread. For Pomona, the sample variance of the air quality index is 100.42, reflecting the degree to which readings diverge from the mean of 48.33. The standard deviation, a complementary measure shown as the square root of the variance, results in a practical calculation: 10.02 for Pomona. It's important to note that these statistics illustrate different elements of air quality. Lower variance and standard deviation in Pomona compared to Anaheim indicate that Pomona's air quality was not only cleaner on average but also more consistent, which can be crucial for residents and policymakers in understanding and improving local air conditions.

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