Answer:
To solve this problem, we'll use the binomial distribution since we're dealing with a random sample of 80 18-20 year olds, and we want to calculate probabilities related to the number of individuals who consumed alcoholic beverages. In this context:
(a) The standard deviation for a binomial distribution is given by the formula:
Standard Deviation (σ) = sqrt(n * p * q)
Where:
- n is the sample size (80 in this case).
- p is the probability of success (the proportion of 18-20 year olds who consumed alcoholic beverages, which is 0.79 in decimal form).
- q is the probability of failure (1 - p).
Let's calculate it:
p = 0.79
q = 1 - p = 1 - 0.79 = 0.21
n = 80
σ = sqrt(80 * 0.79 * 0.21)
σ ≈ 3.07 (rounded to two decimal places)
So, the standard deviation is approximately 3.07.
(b) To find the probability that 75 or more people in this sample have consumed alcoholic beverages, we need to use the binomial probability formula:
P(X ≥ 75) = 1 - P(X < 75)
Where X is the number of people who have consumed alcoholic beverages in the sample.
Now, let's calculate this probability:
P(X < 75) = Σ [from k = 0 to 74] C(80, k) * p^k * q^(80-k)
You'll need to calculate this sum, which can be quite tedious by hand. You can use a calculator or statistical software to find the cumulative probability for X < 75.
Once you find P(X < 75), subtract it from 1 to get P(X ≥ 75).
(c) To find the z-score for a specific value of X, you can use the formula:
z = (X - μ) / σ
Where:
- X is the value you're interested in (in this case, 75).
- μ is the mean of the distribution (μ = n * p).
- σ is the standard deviation we calculated in part (a).
μ = 80 * 0.79 = 63.2 (rounded to one decimal place)
Now, you can calculate the z-score:
z = (75 - 63.2) / 3.07 ≈ 3.84 (rounded to two decimal places)
So, the z-score for 75 people in the sample having consumed alcoholic beverages is approximately 3.84.
Explanation: