According to a survey of American households, the probability that the residents own 2 cars if annual household income is over $50,000 is 80%. Of the households surveyed, 60% had incomes over $50,000 and 70% had 2 cars. The probability that annual household income is over $50,000 if the residents of a household do not own 2 cars is

Question

According to a survey of American households, the probability that the residents own 2 cars if annual household income is over $50,000 is 80%. Of the households surveyed, 60% had incomes over $50,000 and 70% had 2 cars. The probability that annual household income is over $50,000 if the residents of a household do not own 2 cars is

Answer 1

Briarwood Neighborhood Association would like to know how many residents are in favor of a new walking trail. A written survey is distributed to all residents of the subdivision. Then, 45 of the surveys are randomly selected as a sample. 82% of the residents are in favor of the new trail. Which TWO statements are correct for this situation?

Responses
A The population includes all residents of the subdivision.The population includes all residents of the subdivision.
B The population includes only residents who favor the walking trail.The population includes only residents who favor the walking trail.
C The population includes only residents who will use the walking trail.The population includes only residents who will use the walking trail.
D The population includes only the 45 randomly selected surveys.The population includes only the 45 randomly selected surveys.
E "82% of the residents are in favor" is a statistic.

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Answer 2

A and E are the correct statements.

A is correct because the population includes all residents of the subdivision, as they are the group that the association would like to know about.

E is correct because "82% of the residents are in favor" is a sample statistic, as it applies only to the 45 surveys that were randomly selected and not to the entire population.

Answer 3

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Answer 4

To find the probability that annual household income is over $50,000 if the residents of a household do not own 2 cars, we can use Bayes' theorem.

Let's define the events:
A: The residents own 2 cars.
B: Annual household income is over $50,000.

According to the information given:
P(A|B) = 0.80 (Probability of owning 2 cars given an annual household income over $50,000)
P(A) = 0.70 (Probability of owning 2 cars)
P(B) = 0.60 (Probability of annual household income over $50,000)

We want to find P(B|not A), the probability of annual household income over $50,000 given that the residents do not own 2 cars.

We can use Bayes' theorem formula:

P(B|not A) = (P(not A|B) * P(B)) / P(not A)

To find P(not A|B), which is the probability of not owning 2 cars given an annual household income over $50,000, we can apply the complement rule.

P(not A|B) = 1 - P(A|B)

Now let's calculate the values:

P(not A|B) = 1 - P(A|B) = 1 - 0.80 = 0.20

Since the probability of not owning 2 cars given an annual household income over $50,000 is 0.20, we can substitute the values into Bayes' theorem:

P(B|not A) = (P(not A|B) * P(B)) / P(not A)
P(B|not A) = (0.20 * 0.60) / (1 - 0.70) = 0.12 / 0.30
P(B|not A) = 0.40

Therefore, the probability that annual household income is over $50,000 if the residents of a household do not own 2 cars is 0.40 or 40%.