Suppose 60% of adults have a college degree. If I take a random sample of 80 adults, what is the probability at least 50% of them have a college degree?
Use the normal approximation to the binomial distribution.
mean = np = 80 * .6 = ?
standard deviation = √npq = √(80 * .6 * .4) = ?
Calculate. (Note: q = 1 - p)
Next use z-scores:
z = (x - mean)/sd
x = 80 * .5 = ?
Once you have the z-score, find the probability using the z-table.
I hope this will help get you started.
To find the probability that at least 50% of the adults in the sample have a college degree, we can use the binomial distribution.
The binomial distribution calculates the probability of obtaining a certain number of successes (in this case, adults with a college degree) in a fixed number of trials (sample size) with a known probability of success (percentage of adults with a college degree in the population).
In this case, the probability of an adult having a college degree is 60% (or 0.6), and the sample size is 80. We need to calculate the probability of getting 50%, 51%, 52%, and so on up to 80% or 100% of adults with a college degree in the sample.
To find this probability, we can use the cumulative binomial probability formula:
P(X ≥ k) = Σ [C(n, k) * p^k * (1-p)^(n-k)]
Where:
P(X ≥ k) is the probability of getting at least k successes
n is the sample size
k is the number of successes
C(n, k) is the number of combinations of choosing k successes from n trials
p is the probability of success in a single trial
Using this formula, we can calculate the probability of at least 50% of adults having a college degree in the sample.
Alternatively, you can use statistical software or an online calculator that has the capability to calculate cumulative binomial probabilities. By entering the values of n (80), p (0.6), and the desired range (from 50 to 80), the calculator will provide you with the cumulative probability.
Keep in mind that calculating cumulative probabilities for a large range of values manually can get tedious. Using software or an online calculator is usually more efficient.