Problem 89 - Applied Statistics and Probability for Engineers

Question

An article in Atmospheric Chemistry and Physics "Relationship Between Particulate Matter and Childhood AsthmaBasis of a Future Warning System for Central Phoenix" (2012, Vol. \(12,\) pp. \(2479-2490)]\) reported the use of PM10 (particulate matter \(<10 \mu \mathrm{m}\) diameter ) air quality data measured hourly from sensors in Phoenix, Arizona. The 24 -hour (daily) mean PM10 for a centrally located sensor was \(50.9 \mu \mathrm{g} / \mathrm{m}^{3}\) with a standard deviation of \(25.0 .\) Assume that the daily mean of \(\mathrm{PM} 10\) is normally distributed. (a) What is the probability of a daily mean of PM10 greater than \(100 \mu \mathrm{g} / \mathrm{m}^{3} ?\) (b) What is the probability of a daily mean of PM10 less than \(25 \mu \mathrm{g} / \mathrm{m}^{3} ?\) (c) What daily mean of PM10 value is exceeded with probability \(5 \% ?\)

Step-by-Step Solution

Answer

(a) 0.025, (b) 0.150, (c) 92.025 μg/m³

Step 1: Identify given values and distribution

We are given a normally distributed variable, PM10, with a mean \( \mu = 50.9 \, \mu g/m^3 \) and a standard deviation \( \sigma = 25.0 \, \mu g/m^3 \). We need to find probabilities related to this normal distribution.

Step 2: Convert to standard normal variable for part (a)

To find the probability that the daily mean of PM10 is greater than \( 100 \, \mu g/m^3 \), calculate the z-score:\[z = \frac{X - \mu}{\sigma} = \frac{100 - 50.9}{25} = \frac{49.1}{25} = 1.964. \]Then, find the probability \( P(Z > 1.964) \).

Step 3: Calculate probability using standard normal distribution for part (a)

Using a standard normal table, or calculator, we find \( P(Z > 1.964) = 1 - P(Z < 1.964) \). For \( Z = 1.964 \), \( P(Z < 1.964) \approx 0.975 \), so:\[P(Z > 1.964) = 1 - 0.975 = 0.025.\]

Step 4: Convert to standard normal variable for part (b)

To find the probability that the daily mean of PM10 is less than \( 25 \, \mu g/m^3 \), calculate the z-score:\[z = \frac{25 - 50.9}{25} = \frac{-25.9}{25} = -1.036. \]

Step 5: Calculate probability using standard normal distribution for part (b)

Using a standard normal table, or calculator, we find \( P(Z < -1.036) \). For \( Z = -1.036 \), \( P(Z < -1.036) \approx 0.150 \).

Step 6: Find z-value for 5% probability for part (c)

We need to find the PM10 value that is exceeded with a probability of \( 5\% \). This corresponds to the 95th percentile of the normal distribution. From a standard normal distribution table, the z-value for the 95th percentile is approximately \( z = 1.645 \).

Step 7: Convert z-value back to PM10 value for part (c)

Use the z-score formula to find the PM10 value:\[X = \mu + z \sigma = 50.9 + 1.645 \times 25 = 50.9 + 41.125 = 92.025. \]

Key Concepts

Standard Normal Distribution

Let's begin by understanding what standard normal distribution is all about. A standard normal distribution is a special case of the normal distribution that has a mean of zero and a standard deviation of one. In other words, it is a normal distribution that has been standardized. This is achieved through a process called standardization, which involves converting the values to z-scores. Standard normal distribution is widely used because:

• It allows for easier calculation of probabilities.
• It provides a common scale to compare different data sets.

For any normally distributed data set, you can convert it to a standard normal distribution by transforming each point into a z-score. Once transformed, you can utilize standard normal distribution tables or statistical software to calculate probabilities over the data. This standardization simplifies the core calculations associated with assessing probabilities.

Z-Score Calculation

The z-score is a value that represents how many standard deviations a particular value is from the mean. calculating the z-score transforms raw data into an understandable and comparable format, enabling the exploration of its distribution properties.To calculate the z-score, you use the formula:\[ z = \frac{X - \mu}{\sigma} \]where:

• \( X \) is the value of interest,
• \( \mu \) is the mean of the data,
• \( \sigma \) is the standard deviation.

For example, in the exercise, the z-score calculation was used to determine the probability of PM10 levels exceeding or being less than certain values. Once converted into a z-score, the value can be used to look up probabilities from a standard normal distribution table, giving insights into how likely certain events are compared to the overall data spread.

Probability Calculation

Once we have the z-scores, the next step is to calculate probabilities. This is done using standard normal distribution tables, also known as z-tables, or by employing statistical software. Let's break this down:

• For positive z-scores, you look up the value in the z-table to find the probability that a standard normal variable is less than the z-score.
• For negative z-scores, the probability obtained from the z-table tells us the chance that a standard normal variable is less than the z-score.
• To find the probability of being greater than a z-score, subtract the table value from 1.
• For practical applications, such as the PM10 levels, this calculation helps in determining how extreme certain air quality events are compared to historical data.

These probability calculations form a crucial component in the understanding of normal distribution and are particularly useful in environmental statistics and risk assessment.

Environmental Statistics

Environmental statistics involves applying statistical theories to environmental science, providing insights into environmental processes. The use of statistics helps in understanding distributions, such as the particulate matter mentioned in the problem. In this context, environmental statisticians may work with data related to:

• Air quality measurements (like PM10 levels),
• Pollution levels,
• Weather patterns,
• Ecological and wildlife statistics.

By understanding the normal distribution of environmental data, scientists can make predictions about future events. For instance, the calculation of PM10 levels exceeding certain thresholds can inform pollution warnings and public health interventions. Environmental statistics thus play a pivotal role in the monitoring and management of ecological health on both local and global scales.