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 adult's opinion on climate change can be seen as a trial, where 'success' is defined as an adult believing that human activity contributes significantly to climate change. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).
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Probability Mass Function (PMF)
The probability mass function gives the probability of obtaining exactly k successes in n trials for a binomial distribution. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' represents the binomial coefficient. This function is essential for determining the likelihood of observing a specific number of successes, such as exactly 12 adults in this scenario.
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Normal Approximation to the Binomial
For large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. This approximation is valid when both np and n(1-p) are greater than 5. In this case, with n = 25 and p = 0.49, the normal approximation can be used to estimate probabilities, making it easier to analyze the distribution of opinions among the selected adults.
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