Question

College Graduates From the 2010 US Census, we learn that \(27.5 \%\) of US adults have graduated from college. If we take a random sample of 12 US adults, what is the probability that exactly 6 of them are college graduates?

Short answer

Therefore, the probability that exactly 6 out of 12 US adults sampled are college graduates is approximately 0.225.

Step-by-step solution

Determine the Parameters for the Binomial Distribution

For this binomial distribution, the parameters are: number of trials \( n = 12 \), the number of successful trials \( k = 6 \), and the probability of success on a single trial \( p = 0.275 \).

Use the Binomial Probability Formula

The binomial probability formula can be used to calculate the probability of getting exactly 6 successes, it is given by: \[ P(x=k) = C(n, k) * (p^k) * (1-p)^{n-k} \] Where \( C(n, k) \) is the binomial coefficient, which tells us how many ways we can arrange \( k \) successes out of \( n \) trials.

Calculate the Binomial Coefficient

Calculate the number of ways 6 successes can be arranged in a sample of 12. This can be calculated as follows: \[ C(n, k) = \frac{n!}{k! (n-k)!} = \frac{12!}{6! * (12-6)!} = 924 \]

Plug the Values into the Formula

Substitute the given values into the binomial probability formula and solve. \[ P(x=6) = C(12, 6) * (0.275^6) * (1-0.275)^{12-6} = 924 * (0.275^6) * (0.725)^6 \]

Finalize the Calculation

Performing the above calculation yields: \[ P(x=6) \approx 0.225 \]