Forty-nine percent of U.S. adults think that human activity su...

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

Welcome back, everyone. A recent food safety study found that 45% of consumers believe that eating too much processed food significantly harms health. You randomly select 20 consumers. Number one, find the probability that fewer than 2 of them think that eating too much processed food significantly harms health, and number 2, is this event unusual. Let's begin with the first part of this problem, and let's understand that we're dealing with a binomial experiment because we have a fixed number of trials, we are selecting 20 consumers, so this is our number of trials. We have two possible outcomes. Success or failure, our probability of success is 45% or 0.45%, and our selections are independent. So what we're going to do is simply identify the probability that our random variable X is less than 2, because it says fewer than 2 of them think that eating too much processed food significantly harms health. And this would be the probability of X of 0, plus the probability that X is 1, according to the sum rule, right? We have 4 events. So now let's apply the formula for each term, and let's recall that the probability formula says that the probability of the random variable X being equal to the lowercase x is n choose X multiplied by P raises the power of X, multiplied by 1 minus P or raise the power of N minus X. So for the first term, that'll be 22. 0 Multiplied by 0.45, raise to the power of 0. Multiplied by 1 minus 0.45. Raise the power of 20 minus 0. And for the second term, we get 20, choose 1, multiplied by 0.45, raise the power of 1. Multiplied by 1 minus 0.45 raise to the power of 20 minus 1. Or simply 20 choose 0 multiplied by 1 because we're raising to the power of 0. Multiplied by 0.55, raise the power of 20. Plus 2 choose one. Multiplied by 0.45 multiplied by 0.55. To the power of 19. Now 2020 is going to be equal to 20 factorial. Divided by 0 factorial. And also divided by 20 minus 0 factorial. So that's 20 factorial divided by 0 factorial, divided by 20 factorial, which is simply 1, right? So we have 0.55 raise to the power of 20. And now 20 choose one according to the same formula is 20 factorial divided by 19 factorial. Which is 20. And we're multiplying that by 0.45. And 0.55 to the power of 19. Performing the calculation, we get about 0.0001. So that'd be our probability. And for the second part, let's recall that our probability is unusual if it is less than 0.05. In this problem, our probability is 0.0001, and it is indeed less than 0.05, meaning this event is unusual. Well then we have our answers. Let's label them, and thank you for watching.

Key Concepts

Binomial Distribution

Probability Calculation

Unusual Events

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