Question

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 \japanese american and smokes 21 to 30 cigarettes per day.\
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: given that the person is japanese american, the person smokes 21 to 30 cigarettes.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person does not meet either criteria: neither japanese american nor smokes 21 to 30 cigarettes.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: given that the person smokes 21 to 30 cigarettes, the person is japanese american.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: both japanese american and smokes 21 to 30 cigarettes.
○ to pick one person from the study who is japanese american and smokes 21 to 30 cigarettes per day means that the person has to meet at least one of the criteria: either japanese american or smokes 21 to 30 cigarettes.
find the probability (enter your probability as a fraction.)

Explanation:

Part 1: Interpreting "Japanese American AND smokes 21 to 30 cigarettes per day"

The word "AND" in probability (and set theory) means that both conditions must be satisfied. So a person who is "Japanese American AND smokes 21 to 30 cigarettes per day" must be both Japanese American and smoke 21 - 30 cigarettes per day. Let's analyze each option:

• Option 1: Talks about "given that the person is Japanese American", which is a conditional probability (|), not "AND". Eliminate.
• Option 2: Says "does not meet either criteria", which is the opposite of "AND". Eliminate.
• Option 3: Talks about "given that the person smokes 21 to 30 cigarettes", which is a conditional probability (|), not "AND". Eliminate.
• Option 4: Says "has to meet both criteria: both Japanese American and smokes 21 to 30 cigarettes". This matches the definition of "AND" (both conditions must hold).
• Option 5: Says "meet at least one of the criteria", which is the definition of "OR", not "AND". Eliminate.

Answer:

D. To pick one person from the study who is Japanese American and smokes 21 to 30 cigarettes per day means that the person has to meet both criteria: both Japanese American and smokes 21 to 30 cigarettes.

Part 2: Finding the Probability

First, we need to find the total number of people in the study and the number of people who are Japanese American and smoke 21 - 30 cigarettes per day.

Step 1: Calculate the number of people in each smoking category:

• At most 10 cigarettes (≤10): \( 9,886 + 2,745 + 12,831 + 8,378 + 7,650 \)

\( = 9,886 + 2,745 = 12,631 \); \( 12,631 + 12,831 = 25,462 \); \( 25,462 + 8,378 = 33,840 \); \( 33,840 + 7,650 = 41,490 \)

• 11 - 20 cigarettes: \( 6,514 + 3,062 + 4,932 + 10,680 + 9,877 \)

\( = 6,514 + 3,062 = 9,576 \); \( 9,576 + 4,932 = 14,508 \); \( 14,508 + 10,680 = 25,188 \); \( 25,188 + 9,877 = 35,065 \)

• 21 - 30 cigarettes: \( 1,671 + 1,419 + 1,406 + 4,715 + 6,062 \)

\( = 1,671 + 1,419 = 3,090 \); \( 3,090 + 1,406 = 4,496 \); \( 4,496 + 4,715 = 9,211 \); \( 9,211 + 6,062 = 15,273 \)

• At least 31 cigarettes (≥31): \( 759 + 788 + 800 + 2,305 + 3,970 \)

\( = 759 + 788 = 1,547 \); \( 1,547 + 800 = 2,347 \); \( 2,347 + 2,305 = 4,652 \); \( 4,652 + 3,970 = 8,622 \)

Step 2: Total number of people (\( N \)):

\( N = 41,490 + 35,065 + 15,273 + 8,622 \)
\( 41,490 + 35,065 = 76,555 \); \( 76,555 + 15,273 = 91,828 \); \( 91,828 + 8,622 = 100,450 \)

Step 3: Number of Japanese Americans who smoke 21 - 30 cigarettes per day (\( n \)):

From the "21 - 30 cigarettes" category, the number of Japanese Americans is \( 4,715 \).

Step 4: Calculate the probability (\( P \)):

Probability is the number of favorable outcomes (Japanese American and 21 - 30 cigarettes) divided by the total number of outcomes (total people). So:
\( P = \frac{\text{Number of Japanese Americans who smoke 21 - 30 cigarettes}}{\text{Total number of people}} = \frac{4,715}{100,450} \)
Simplify the fraction by dividing numerator and denominator by 5: \( \frac{4,715 \div 5}{100,450 \div 5} = \frac{943}{20,090} \) (we can check if 943 and 20,090 have a common factor. 943 ÷ 23 = 41, 20,090 ÷ 23 = 873.478... No, so 943/20090 is simplified? Wait, wait, let's re - check the total:

Wait, let's recalculate the total number of people:

• ≤10: 9,886 (African Americans) + 2,745 (Native Hawaiians) + 12,831 (Latinos) + 8,378 (Japanese Americans) + 7,650 (Whites)

9,886 + 2,745 = 12,631; 12,631 + 12,831 = 25,462; 25,462 + 8,378 = 33,840; 33,840 + 7,650 = 41,490. Correct.

• 11 - 20: 6,514 + 3,062 + 4,932 + 10,680 + 9,877

6,514 + 3,062 = 9,576; 9,576 + 4,932 = 14,508; 14,508 + 10,680 = 25,188; 25,188 + 9,877 = 35,065. Correct.

• 21 - 30: 1,671 + 1,419 + 1,406 + 4,715 + 6,062

1,671 + 1,419 = 3,090; 3,090 + 1,406 = 4,496; 4,496 + 4,715 = 9,211; 9,211 + 6,062 = 15,273. Correct.

• ≥31: 759 + 788 + 800 + 2,305 + 3,970

759 + 788 = 1,547; 1,547 + 800 = 2,347; 2,347 + 2,305 = 4,652; 4,652 + 3,970 = 8,622. Correct.

Total: 41,490 + 35,065 = 76,555; 76,555 + 15,273 = 91,828; 91,828 + 8,622 = 100,450. Correct.

Number of Japanese Americans who smoke 21 - 30: 4,715. Correct.

Now, simplify \( \frac{4715}{100450} \). Let's divide numerator and denominator by 5: 4715 ÷ 5 = 943; 100450 ÷ 5 = 20090. Now, check if 943 and 20090 have a common divisor. Let's check 943: 943 ÷ 23 = 41 (23×41 = 943). Now check 20090 ÷ 23: 23×873 = 20079, 20090 - 20079 = 11, so 20090 is not divisible by 23. So the simplified fraction is \( \frac{943}{20090} \). Wait, but let's check the arithmetic again for the number of Japanese Americans in each category:

Wait, in the "at most 10" category, Japanese Americans: 8,378; "11 - 20": 10,680; "21 - 30": 4,715; "≥31": 2,305. But for our problem, we only care about "21 - 30" and Japanese American, which is 4,715. Total people: 100,450. So the probability is \( \frac{4715}{100450}=\frac{943}{20090} \) (or we can write it as \( \frac{4715}{100450} \) if we don't simplify, but simplifying is better).

Step - by - Step Format:

Step1: Calculate total number of people

First, sum the number of people in each smoking group:

• At most 10 cigarettes: \( 9886 + 2745 + 12831 + 8378 + 7650 = 41490 \)
• 11 - 20 cigarettes: \( 6514 + 3062 + 4932 + 10680 + 9877 = 35065 \)
• 21 - 30 cigarettes: \( 1671 + 1419 + 1406 + 4715 + 6062 = 15273 \)
• At least 31 cigarettes: \( 759 + 788 + 800 + 2305 + 3970 = 8622 \)

Total number of people: \( 41490 + 35065 + 15273 + 8622 = 100450 \)

Step2: Identify favorable outcomes

Number of Japanese Americans who smoke 21 - 30 cigarettes per day: \( 4715 \)

Step3: Calculate probability

Probability \( P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{4715}{100450} \)

Step4: Simplify the fraction

Divide numerator and denominator by 5: \( \frac{4715\div5}{100450\div5}=\frac{943}{20090} \)