Question

Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (c) less than two. (d) Are any of these events unusual? Explain your reasoning.

Accepted Answer

Step 1: Identify the type of probability distribution. Since the problem involves a fixed number of trials (25 adults), each with two possible outcomes (think human activity contributes a great deal or not), and the probability of success (49%) is constant, this is a binomial distribution. The number of trials (n) is 25, and the probability of success (p) is 0.49. Step 2: Define the random variable X. Let X represent the number of adults who think human activity contributes a great deal to climate change. The goal is to find P(X \u003c 2), which is the probability that fewer than 2 adults think this way. Step 3: Use the binomial probability formula to calculate P(X = 0) and P(X = 1). The formula is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' is the binomial coefficient. For X = 0 and X = 1, substitute the values of n, k, and p into the formula. Step 4: Add the probabilities for P(X = 0) and P(X = 1) to find P(X \u003c 2). Since 'less than 2' includes both X = 0 and X = 1, sum the probabilities calculated in the previous step. Step 5: To determine if the event is unusual, compare the probability P(X \u003c 2) to a threshold (typically 0.05). If P(X \u003c 2) is less than 0.05, the event is considered unusual. Explain your reasoning based on this comparison.

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Concetti chiave

Binomial Distribution

Probability Calculation

Unusual Events