Question

The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.8% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status.

Construct the probability distribution function (PDF). (Round your probabilities to five decimal places.)

x	P(x)
0
1
2
3
4
5
6
7
8

What is the standard deviation (σ)? (Round your answer to four decimal places.)
σ =

What is the probability that at least two of the freshmen reply "yes"? (Round your answer to four decimal places.)

Answer 1

Answer 2

To construct the probability distribution function (PDF), we can use the binomial distribution since we are interested in the number of students who believe that same-sex couples should have the right to legal marital status.

The given information allows us to determine the probability of success, which is the probability that an individual student replies "yes." From the data, 71.8% of the students replied "yes." Therefore, the probability of success is: p = 0.718

The number of trials is 8, as we randomly pick eight freshmen.

Using these values, we can calculate the PDF for each possible number of students who replied "yes" (x) using the binomial distribution formula:

P(x) = C(n, x) * p^x * (1 - p)^(n - x)

Where: n = number of trials (8) x = number of successes (0 to 8) p = probability of success (0.718) C(n, x) = combination function (n choose x)

Now let's calculate the PDF:

x P(x) 0 0.00172678 1 0.01793343 2 0.08721772 3 0.21228432 4 0.31807580 5 0.26716236 6 0.12035669 7 0.02521827 8 0.00203553

To calculate the standard deviation (σ) of the distribution, we can use the formula for the binomial distribution:

σ = √(n * p * (1 - p))

σ = √(8 * 0.718 * (1 - 0.718))

σ ≈ 1.3322

Therefore, the standard deviation (σ) is approximately 1.3322.

To find the probability that at least two of the freshmen reply "yes," we sum the probabilities for x ≥ 2:

P(at least two "yes") = P(2) + P(3) + P(4) + P(5) + P(6) + P(7) + P(8)

P(at least two "yes") ≈ 0.8774

Therefore, the probability that at least two of the freshmen reply "yes" is approximately 0.8774