Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, each selected adult either believes that human activity contributes to climate change (success) or does not (failure). The parameters of the distribution are the number of trials (n = 25) and the probability of success (p = 0.49).
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Probability Mass Function (PMF)
The probability mass function gives the probability of obtaining a specific number of successes in a binomial distribution. For a given number of successes k, the PMF is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k). This function is essential for determining the probability of observing between 8 and 11 adults who believe in the contribution of human activity to climate change.
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Cumulative Probability
Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain number. To find the probability that the number of adults believing in human activity's contribution to climate change is between 8 and 11, one would calculate the cumulative probabilities for 11 and 7, then subtract the two. This approach allows for the determination of the probability of a range of outcomes.
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