An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate P ( 45 − 54 years old ) by dividing the number of respondents in the 45-54 age group by the total number of respondents: 2143 540 ≈ 0.252 .
Calculate P ( 45 − 54 years old ∣ more likely ) by dividing the number of respondents in the 45-54 age group who answered 'More likely' by the total number of respondents who answered 'More likely': 1322 359 ≈ 0.272 .
Check if the events are independent by comparing P ( 45 − 54 years old ) and P ( 45 − 54 years old ∣ more likely ) . Since 0.252 = 0.272 , the events are not independent.
The events '45-54 years old' and 'more likely' are not independent. P ( 45 − 54 years old ) = 0.252 , P ( 45 − 54 years old ∣ more likely ) = 0.272 , Not independent
Explanation
Analyze the problem and data We are given a contingency table from a Harris Poll that surveyed adult Americans about their likelihood to buy products 'Made in America'. The table breaks down the responses by age group. We need to find P ( 45 − 54 years old ) , P ( 45 − 54 years old ∣ more likely ) , and determine if the events '45-54 years old' and 'more likely' are independent.
Calculate P(45-54 years old) First, we need to calculate P ( 45 − 54 years old ) . This is the probability that a randomly selected respondent is in the 45-54 age group. We can calculate this by dividing the number of respondents in the 45-54 age group by the total number of respondents. From the table, we see that there are 540 respondents in the 45-54 age group, and the total number of respondents is 2143. Therefore, P ( 45 − 54 years old ) = 2143 540 ≈ 0.252
Calculate P(45-54 years old | more likely) Next, we need to calculate P ( 45 − 54 years old ∣ more likely ) . This is the conditional probability that a respondent is in the 45-54 age group, given that they answered 'More likely'. We can calculate this by dividing the number of respondents in the 45-54 age group who answered 'More likely' by the total number of respondents who answered 'More likely'. From the table, we see that there are 359 respondents in the 45-54 age group who answered 'More likely', and the total number of respondents who answered 'More likely' is 1322. Therefore, P ( 45 − 54 years old ∣ more likely ) = 1322 359 ≈ 0.272
Determine independence Finally, we need to determine if the events '45-54 years old' and 'more likely' are independent. Two events are independent if and only if P ( A ∣ B ) = P ( A ) . In our case, we need to check if P ( 45 − 54 years old ∣ more likely ) = P ( 45 − 54 years old ) . We have already calculated these probabilities: P ( 45 − 54 years old ) ≈ 0.252 P ( 45 − 54 years old ∣ more likely ) ≈ 0.272 Since 0.272 = 0.252 , the events are not independent.
Alternative method to determine independence Alternatively, we can check if P ( 45 − 54 years old ∩ more likely ) = P ( 45 − 54 years old ) ⋅ P ( more likely ) .
P ( 45 − 54 years old ∩ more likely ) = 2143 359 ≈ 0.1675 P ( 45 − 54 years old ) ⋅ P ( more likely ) = 2143 540 ⋅ 2143 1322 ≈ 0.1554 Since 0.1675 = 0.1554 , the events are not independent.
State the final answer Therefore, P ( 45 − 54 years old ) = 2143 540 ≈ 0.252 , P ( 45 − 54 years old ∣ more likely ) = 1322 359 ≈ 0.272 , and the events '45-54 years old' and 'more likely' are not independent.
Examples
Contingency tables and probability calculations are used extensively in market research to analyze customer behavior and preferences. For example, a company might use a survey to understand the relationship between age groups and their likelihood to purchase a new product. By calculating conditional probabilities, the company can target specific age groups with tailored marketing campaigns, increasing the effectiveness of their advertising efforts. This targeted approach ensures that marketing resources are used efficiently and that the right message reaches the right audience.
By calculating the total charge from the current of 15.0 A over 30 seconds, we find that 450 coulombs of charge pass through. Dividing this by the charge of a single electron (approximately 1.602 x 10^-19 C), we find that around 2.81 x 10^21 electrons flow through the device. Thus, approximately 2.81 billion billion electrons are involved.
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