According to the statistics of a famous high school in Canada, all their graduates went to either A or B for university study. Following are the proportions of graduates who went to the two universities.
University Proportion
A 1-k
B k
Suppose k = 0.4
(a) What is the standard deviation of the number of graduates going to B from random samples of the graduates, each with sample size 50?
(b) What is the probability that more than half of graduates going to A in a random sample of 8 graduates?
(c) What is the chance of having at least one graduate going to B and at least one graduate going to A in a random sample of 10 graduates?
(d) Based on the most updated annual report from the school, there were 1280 graduates last year and 690 of them went to A. What is the estimated value of k?
(a) To find the standard deviation of the number of graduates going to B, we need to use the formula for the standard deviation of a binomial distribution:
σ = √(N * p * q)
where σ is the standard deviation, N is the sample size, p is the proportion of success (in this case, the proportion of graduates going to B), and q is the proportion of failure (1 - p).
In this case, N = 50 and p = k = 0.4. Therefore:
q = 1 - p = 1 - 0.4 = 0.6
σ = √(50 * 0.4 * 0.6) = √(12) = 3.464
So the standard deviation of the number of graduates going to B in random samples of 50 graduates is approximately 3.464.
(b) To find the probability that more than half of the graduates going to A in a random sample of 8 graduates, we need to use the binomial distribution formula:
P(X > n) = 1 - P(X ≤ n)
where X is the random variable representing the number of graduates going to A, and n is the desired number of successes.
In this case, X follows a binomial distribution with parameters n = 8 and p = 1 - k = 0.6 (the proportion of graduates going to A). We want to find P(X > 4) (more than half of the graduates going to A).
Using a binomial probability calculator or table, we find:
P(X > 4) = 1 - P(X ≤ 4)
P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
P(X > 4) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))
Calculating these probabilities using the binomial distribution formula:
P(X > 4) ≈ 1 - (0.0192 + 0.1106 + 0.2756 + 0.3459 + 0.2306)
P(X > 4) ≈ 1 - 0.9819
P(X > 4) ≈ 0.0181
So the probability that more than half of the graduates going to A in a random sample of 8 graduates is approximately 0.0181.
(c) To find the chance of having at least one graduate going to B and at least one graduate going to A in a random sample of 10 graduates, we can calculate the complementary probability of having no graduates going to B or A.
Since all the graduates go to either A or B, if there are no graduates going to B or A, it means all the graduates went to one university only. Therefore:
P(at least one graduate going to B and A) = 1 - P(no graduates going to B or A)
P(no graduates going to B or A) = P(all graduates going to A) + P(all graduates going to B)
Since P(all graduates going to A) = p^N and P(all graduates going to B) = q^N, where N is the sample size and p and q are the proportions of graduates going to A and B respectively.
In this case, N = 10, p = 1 - k = 0.6, and q = k = 0.4.
P(no graduates going to B or A) = 0.6^10 + 0.4^10
P(at least one graduate going to B and A) = 1 - (0.6^10 + 0.4^10)
Calculating this probability:
P(at least one graduate going to B and A) ≈ 1 - (0.000104 + 0.000104)
P(at least one graduate going to B and A) ≈ 1 - 0.000208
P(at least one graduate going to B and A) ≈ 0.999792
So the chance of having at least one graduate going to B and at least one graduate going to A in a random sample of 10 graduates is approximately 0.999792.
(d) To estimate the value of k, we can use the proportion of graduates going to A from the sample of 1280 graduates.
Given that there were 690 graduates going to A, we can calculate the estimated value of k as:
k = number of graduates going to B / total number of graduates
k = (1280 - 690) / 1280 = 590 / 1280 ≈ 0.4609
Therefore, the estimated value of k is approximately 0.4609.