if the true proportion of californians who are unemployed is 11%, what is the probability that a sample of 100 californians are unemployed?
To find the probability that a sample of 100 Californians is unemployed given that the true proportion of unemployed Californians is 11%, you can use the binomial distribution formula.
The binomial distribution is used to calculate the probability of a specific number of successes (unemployed individuals) in a fixed number of independent trials (the sample size). In this case, the number of successes is the number of unemployed Californians in the sample, and the total number of trials is the sample size (100).
The formula for the probability of k successes in n trials, where p is the probability of success on a single trial, is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case, the probability of success (p) is the true proportion of unemployed Californians (11% or 0.11), and the sample size (n) is 100.
Using these values in the formula, the probability that a sample of 100 Californians are unemployed can be calculated as follows:
P(X = k) = (100 choose k) * 0.11^k * (1 - 0.11)^(100 - k)
To find the probability for a specific value of k, substitute that value into the equation and evaluate it. If you're interested in the probability of any number of unemployed participants in the sample (k = 0, 1, 2, ..., 100), you would need to calculate the probability for each value of k separately.
For example, the probability that exactly 10 Californians are unemployed in the sample of 100 can be calculated as:
P(X = 10) = (100 choose 10) * 0.11^10 * (1 - 0.11)^(100 - 10)
You can repeat this calculation for any desired value of k to find the corresponding probability.