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An article under review for Air Quality, Atmosphere \(\&\) Health titled "Linking Particulate Matter (PM10) and Childhood Asthma in Central Phoenix" linked air quality to childhood asthma incidents. The study region in central Phoenix, Arizona, recorded 10.500 asthma incidents in chil dren in a 21 -month period. Assume that the number of asthma incidents follows a Poisson distribution. (a) Approximate the probability of more than 550 asthma incidents in a month. (b) Approximate the probability of 450 to 550 asthma incidents in a month. (c) Approximate the number of asthma incidents exceeded with probability \(5 \%\). (d) If the number of asthma incidents were greater during the winter than the summer, what would this imply about the Poisson distribution assumption?

Step by step solution

01

Calculate the Monthly Rate of Incidents

The problem states that there are 10,500 asthma incidents over a 21-month period. Calculate the average rate of incidents per month, \( \lambda \), by dividing the total incidents by the number of months. \[ \lambda = \frac{10,500}{21} = 500 \] incidents per month.

02

Probability of More than 550 Incidents

To find the probability of more than 550 incidents, we use the Poisson cumulative distribution function (CDF). The complement of the CDF gives us the probability of more than a certain number of incidents. Calculate \( P(X > 550) \), which is \( 1 - P(X \leq 550) \). Use a Poisson calculator or table for \( \lambda = 500 \).

03

Probability of Incidents Between 450 and 550

To find the probability of incidents falling between 450 and 550, compute \( P(450 \leq X \leq 550) \) using the CDF: \( P(X \leq 550) - P(X < 450) \). Use a Poisson calculator or table to determine these probabilities.

04

Number Exceeded with Probability 5%

This requires finding the 95th percentile of the Poisson distribution with \( \lambda = 500 \). You can use a percentile calculator or table for the Poisson distribution to find the value \( k \) at which \( P(X \leq k) = 0.95 \).

05

Implications of Seasonal Variation

If the incidents are higher in winter than in summer, this could imply the rate \( \lambda \) is not constant over time, violating the Poisson distribution assumption. The Poisson distribution assumes a constant rate of occurrence over time.

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Key Concepts

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

Air Quality and Health

Poor air quality has a significant impact on public health. It can lead to various health issues, including respiratory diseases like asthma, especially in children. Particulate Matter (PM) is one of the common pollutants affecting air quality. PM10 refers to tiny particles suspended in the air that can penetrate the respiratory tract. Exposure to high levels of PM10 can exacerbate asthma symptoms and trigger asthma attacks.

• Air pollution is closely monitored in urban areas to protect vulnerable populations.
• People with pre-existing health conditions like asthma are more susceptible to poor air quality.

Studies, like the one mentioned in central Phoenix, aim to establish a correlation between air quality and health outcomes, allowing authorities to take action such as implementing stricter regulations to improve the air quality and curb health issues.

Childhood Asthma

Childhood asthma is a chronic condition characterized by frequent wheezing, coughing, and difficulty breathing. It is often triggered by environmental factors such as allergens and pollutants like PM10. Areas with poor air quality rates see higher asthma incidents among children.

• Genetics and environmental factors both play roles in childhood asthma.
• It's crucial to monitor asthma incidence rates to better understand the underlying causes.

Educational programs in areas with high asthma rates can help manage symptoms and reduce the frequency of asthma attacks. For instance, teaching families how to minimize indoor and outdoor pollutants' exposure can be an effective way to combat childhood asthma.

Probability Estimation

Probability estimation involves calculating the likelihood of an event or outcome. In the context of the exercise, we use the Poisson distribution to estimate probabilities of monthly asthma incidents.

• The Poisson distribution is suitable for modeling the number of times an event happens within a fixed period.
• It requires knowing the average rate ( \(\lambda \) ) of occurrence.

For example, estimating the probability of having more than 550 asthma incidents in a month involves computing the complement of the probability of having 550 or fewer incidents. This helps public health officials prepare and allocate resources accordingly.

Seasonal Variation Analysis

Seasonal variation occurs when data shows regular seasonal patterns over time. In the context of asthma incidents, this could imply differing incidence rates between seasons like winter and summer.

• Seasonal trends can affect the assumptions of a Poisson distribution, which presumes a constant rate.
• If asthma incidents rise in winter, it indicates a departure from this assumption.

Analyzing seasonal variation is critical in understanding the full scope of asthma incidents. It can influence healthcare planning, leading to more effective interventions. For example, resources might be increased in healthcare facilities during peak seasons to better manage the increased patient load.