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Use the following information to answer the next seven exercises. An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 Whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 Whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 Whites. In words, explain what it means to pick one person from the study who is 鈥淛apanese American OR smokes 21 to 30 cigarettes per day.鈥� Also, find the probability.

Step by step solution

01

Identify the Total Population

First, we need to calculate the total number of people included in the study, regardless of their smoking habits. Sum the number of people from each ethnicity group and smoking level.\[ \text{Total} = (9886 + 2745 + 12831 + 8378 + 7650) + (6514 + 3062 + 4932 + 10680 + 9877) \] \[ + (1671 + 1419 + 1406 + 4715 + 6062) + (759 + 788 + 800 + 2305 + 3970) \] Calculate this to find the total population.

02

Determine "Japanese American" Group

Calculate the number of Japanese Americans in the study across all smoking levels.\[ \text{Japanese Americans} = (8378 + 10680 + 4715 + 2305) \] Sum these values to find the number of Japanese Americans.

03

Determine "21 to 30 Cigarettes per Day" Group

Calculate the people who smoke 21 to 30 cigarettes per day across all ethnicities.\[ \text{21 to 30 cigarettes} = (1671 + 1419 + 1406 + 4715 + 6062) \] Add these values to find the total for this group.

04

Count Both "Japanese American" and "21 to 30 Cigarettes per Day"

To avoid double counting, find the number of Japanese Americans within those who smoke 21 to 30 cigarettes per day. From the data, 4715 Japanese Americans smoke this amount.

05

Apply the Inclusion-Exclusion Principle

Calculate the total number of people who are either "Japanese American" OR smoke "21 to 30 cigarettes per day" using the formula: \[ \text{Total} = (\text{Japanese Americans}) + (\text{21 to 30 cigarettes}) - (\text{Both}) \] Substitute the values from previous steps.

06

Calculate the Probability

Now find the probability of picking a person from the criteria "Japanese American OR smokes 21 to 30 cigarettes per day." The formula is: \[ P(\text{Japanese American OR 21 to 30 cigarettes}) = \frac{\text{Total from Step 5}}{\text{Total population from Step 1}} \] Compute the probability.

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Key Concepts

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

Data Analysis

Data analysis is a powerful tool used to process and interpret the voluminous data collected from studies like this one from the New England Journal of Medicine. It involves inspecting, cleaning, and modeling data with the goal of deriving valuable insights.
In the context of this study, we're examining the self-reported ethnicity and smoking levels of different ethnic groups across California and Hawaii. By analyzing the data, we can start to see patterns, such as which ethnic groups tend to smoke more heavily or lightly.
The first step of data analysis in this exercise is identifying the total number of participants. This involves summing individuals across different ethnicities and smoking levels.

• Total participants who smoke at most ten cigarettes include groups like African Americans, Native Hawaiians, etc.
• This continues through all smoking level categories, culminating in a comprehensive overview of the study's scope.

Once we have the total, we can delve deeper into analyzing subsets of the population, such as those who are Japanese American, or those who smoke a specific range of cigarettes per day.

Statistical Methods

Statistical methods are essential for interpreting collected data, helping us understand trends and probabilities. In this exercise, these methods are used to determine the probability of an event occurring within a study's sample group.
Here, statistical methods help us figure out the probability that a randomly picked person from the study is either Japanese American or smokes 21 to 30 cigarettes per day. To do this, we need to determine:

• The total number of Japanese Americans across all smoking categories.
• The total number of people who smoke 21 to 30 cigarettes.
• The overlap between these two groups, Japanese Americans who smoke 21 to 30 cigarettes.

By using a probability formula, we can calculate the likelihood of selecting a participant fitting these criteria, a fundamental approach in statistics for an objective analysis.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a core concept in probability theory and statistics, used to calculate the probability of either event A or event B occurring by considering their overlap.
In this exercise, we use the inclusion-exclusion principle to identify the total number of individuals who are either Japanese American or smoke 21 to 30 cigarettes per day. This is crucial to avoid double-counting those who fall into both categories.
The calculation involves these steps:

• Add the total number of Japanese Americans to the total number of people who smoke 21 to 30 cigarettes.
• Subtract the number of individuals who fit into both criteria (Japanese Americans smoking 21 to 30 cigarettes).

This systematic approach enables us to find the precise total for people meeting either one of the specified conditions without redundancy, thus ensuring accurate probability measurement.