Assume it is known that the probability of birth is equal in all months. What is the probability
that in the STAT class of 120 students, exactly 20 students have their birthdays in either August or September? Solve using (i) the exact Binomial distribution, (ii) the Normal approximation to the Binomial distribution.
If "the probability of birth is equal in all months," that means that you would expect 10 students to be born in each month. Would either distribution apply in this case?
Have you manged to solve this question ?
To solve this problem using the exact binomial distribution, we need to first calculate the probability of a single student having their birthday in August or September.
Since the probability of birth is equal in all months, and there are 12 months in a year, the probability of a student having their birthday in August or September is 2/12 = 1/6.
Now, we can use the binomial probability formula to find the probability of exactly 20 students out of 120 having their birthdays in either August or September. The formula is as follows:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) represents the probability of having exactly k successes (in this case, students having their birthdays in either August or September),
C(n, k) is the number of ways to choose k items from a set of n items (also known as the binomial coefficient),
p is the probability of a single success (1/6 in this case),
n is the total number of trials (the number of students in the class, 120), and
k is the number of successful trials.
So, in our case, we have:
P(X = 20) = C(120, 20) * (1/6)^20 * (5/6)^(120-20)
To solve this expression, we need to calculate the binomial coefficient C(120, 20) and raise the probabilities to the appropriate powers. This calculation can be performed using a calculator or statistical software.
Now, let's solve the problem using the Normal approximation to the binomial distribution.
When the sample size (n) is large and the probability of success (p) is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution. In this case, n = 120 and p = 1/6, so we can use the normal approximation.
To apply the normal approximation, we need to calculate the mean (μ) and standard deviation (σ) of the binomial distribution:
μ = n * p
σ = sqrt(n * p * (1-p))
Using these formulas, we get:
μ = 120 * (1/6)
σ = sqrt(120 * (1/6) * (5/6))
Now that we have the mean and standard deviation, we can approximate the binomial distribution using the normal distribution. We want to find the probability of exactly 20 students having their birthdays in either August or September.
P(X = 20) ≈ P(19.5 < X < 20.5)
To calculate this probability, we standardize the values using the normal distribution and look up the corresponding area under the standard normal curve using a z-table or a calculator with built-in functions.
By using these methods, you can solve the problem using either the exact binomial distribution or the normal approximation to the binomial distribution.