opinion polls find that 16 percent of americans adults claim that they never have time to relax. suppose you take a random sample of 400 american adults and count the number X in your sample and claim that they never never have time to relax.
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To determine the probability distribution for the number of American adults in your sample who claim they never have time to relax, we can use the binomial distribution.
The binomial distribution is appropriate here because we have a fixed number of trials (the 400 American adults in your sample), and each trial can result in one of two outcomes: claiming to never have time to relax or not claiming to never have time to relax.
Let's break down the problem using the binomial distribution formula:
1. Determine the probability of success (p):
In this case, the probability is given as 16 percent or 0.16. This is the probability of an American adult claiming to never have time to relax.
2. Determine the number of trials (n):
The number of trials is the size of your sample, which is 400.
3. Calculate the probability distribution:
Let X denote the number of American adults in your sample who claim they never have time to relax. The probability of having exactly x successes can be calculated using the binomial probability formula:
P(X = x) = C(n, x) * p^x * (1 - p)^(n - x)
Where C(n, x) denotes the number of combinations of n items taken x at a time, and p is the probability of success.
For example, let's calculate the probability of having exactly 50 American adults in your sample who claim they never have time to relax. Plugging the values into the formula, we get:
P(X = 50) = C(400, 50) * 0.16^50 * (1 - 0.16)^(400 - 50)
You can use a calculator or statistical software to compute the probability (P(X = 50)).
Using this formula, you can calculate the probability distribution for any number of American adults in your sample who claim they never have time to relax, from 0 to 400.
Note that probabilities for individual values will be small, so it's common to calculate ranges of values or use cumulative probabilities to determine the likelihood of a specific outcome.
I hope this explanation helps you understand how to determine the probability distribution for this scenario and calculate the probabilities for different numbers of American adults who claim they never have time to relax in your sample.