Question

72% of all students at a college still need to take another math class. Find the probability for the following given that a sample size of 45 students are randomly selected. Round your answer to four places.

a) Exactly 32 of them need to take another math class.
b) At most 34 of them need to take another math class.
c) At least 33 of them need to take another math class.
d) Between 29 and 36 (inclusive) of them need to take another math class.

Answer 1

To find the probabilities, we need to use the binomial probability formula and calculate the expressions for each case.

Step-by-step explanation:

In order to find the probability in each case, we need to use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where:

n = sample size

k = number of trials with the specified outcome

p = probability of the specified outcome

For the given question:

a) Exactly 32: n = 45, k = 32, p = 0.72

Plugging in the values, we get:

P(X = 32) = (45 choose 32) * 0.72^32 * (1-0.72)^(45-32)

Calculating this expression will give us the probability.

b) At most 34: We need to find P(X ≤ 34). Using the binomial cumulative distribution function, we can find the probability of X being less than or equal to 34.

c) At least 33: We need to find P(X ≥ 33). Again, using the binomial cumulative distribution function, we can find the probability of X being greater than or equal to 33.

d) Between 29 and 36 (inclusive): We need to find P(29 ≤ X ≤ 36). Using the binomial cumulative distribution function, we can find the probability of X being between 29 and 36 (inclusive).