Question

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities
Thirty-six percent of Americans think there is still a need for the practice of changing their clocks for Daylight Savings Time. You randomly select seven Americans. Find the probability that the number who say there is still a need for changing their clocks for Daylight Savings Time is (c) at least six.

Accepted Answer

Step 1: Identify the type of distribution to use. Since the problem involves a fixed number of trials (7 Americans), each with two possible outcomes (agree or disagree), and a constant probability of success (36% or 0.36), this is a binomial distribution problem. Step 2: Define the parameters of the binomial distribution. The number of trials (n) is 7, the probability of success (p) is 0.36, and the number of successes (x) is at least 6. This means we are interested in P(X ≥ 6). Step 3: Use the complement rule to simplify the calculation. P(X ≥ 6) can be rewritten as 1 - P(X ≤ 5). This allows us to calculate the cumulative probability for X ≤ 5 and subtract it from 1. Step 4: Use the binomial probability formula to calculate P(X ≤ 5). The formula for the binomial probability is: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n! / [k! * (n-k)!]. Compute this for k = 0, 1, 2, 3, 4, and 5, then sum these probabilities to find P(X ≤ 5). Step 5: Subtract the cumulative probability P(X ≤ 5) from 1 to find P(X ≥ 6). Finally, determine whether the event is unusual by comparing the probability to a threshold (e.g., 0.05). If P(X ≥ 6) is less than 0.05, the event is considered unusual.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Geometric Distribution

Binomial Distribution

Unusual Events