At a small college it is known from previous years that 35% of all college freshman start in college algebra. The new freshman class has 100 students in it,

What is the probability that between 35 and 42 freshman will start in college algebra?
Is it reasonable to approximate this distribution using a normal curve? Why or why not?
What is the probability that at most 24 students will starting college algebra?
What is the probability that at least 48 students will start in college algebra?
This college tries to kep class size down to 24 studentsfor freshman. Why might the probabilities in the previous 2 questions be important to know?

Sixty percent of the students at a particular community college are female.

If 12 students at that community college are randomly selected,
find the probability that 5 will be female.

To find the probability that between 35 and 42 freshman will start in college algebra, we can use the binomial probability formula.

The binomial probability formula is:

P(x) = (nCx) * (p^x) * ((1-p)^(n-x))

Where:
P(x) is the probability of getting exactly x successes
n is the number of trials or total number of students (100 in this case)
x is the number of successes or students starting in college algebra
p is the probability of success or the proportion of college freshman starting in college algebra (0.35 in this case)
(1-p) is the probability of failure or the proportion of college freshman not starting in college algebra (0.65 in this case)
nCx is the binomial coefficient, calculated as n! / (x! * (n-x)!), which represents the number of ways to choose x objects from a set of n objects

By calculating P(x) for x = 35, 36, 37, 38, 39, 40, 41, 42 and summing up these individual probabilities, we can find the probability that between 35 and 42 freshmen will start in college algebra.

To determine if it is reasonable to approximate this distribution using a normal curve, we need to check if the conditions for using the normal approximation to the binomial distribution are met. These conditions are:
1. The number of trials (n) is large (rule of thumb: n >= 30).
2. Both p (probability of success) and q (probability of failure = 1 - p) are large (rule of thumb: np >= 5 and nq >= 5).

In this case, n = 100, p = 0.35, and q = 0.65. Since np = 100 * 0.35 = 35 and nq = 100 * 0.65 = 65, both np and nq are greater than or equal to 5, so the conditions are met. Therefore, it is reasonable to approximate this distribution using a normal curve.

To find the probability that at most 24 students will start in college algebra, we can sum up the probabilities of x = 0, 1, 2, ..., 24 using the binomial probability formula. Alternatively, since the number of students starting in college algebra follows a binomial distribution, we can use the cumulative distribution function (CDF) of the binomial distribution to find P(x <= 24).

To find the probability that at least 48 students will start in college algebra, we can calculate the complement probability of P(x <= 47). This can be done by subtracting P(x <= 47) from 1.

The probabilities in the previous two questions are important to know because they provide insights into the potential class sizes for the college algebra course. If the probability of having at most 24 students starting in college algebra is high, it suggests that the course may not face overcrowding issues. On the other hand, if the probability of having at least 48 students starting in college algebra is high, it indicates that the college may need to make arrangements to accommodate a larger class size or consider opening additional sections of the course.