Assume that 60% of community college students are female.
1) If we pick 10 students at random justify why we can or cannot model this using a normal model.
2) Find the probability that 8 of the 10 students selected are female.
3) Find the probability that 8 of more of the 10 students selected are female.
1) To determine if we can model the scenario using a normal model, we need to check if the conditions for using the normal distribution are satisfied. The normal distribution can be used when the sample size is large enough, typically greater than 30, and the data follows a bell-shaped curve (approximately symmetric).
In this case, we are selecting 10 students at random. Since the sample size is relatively small, the condition for using a normal model is not satisfied. Additionally, the distribution of gender in the population may not necessarily follow a normal distribution. Therefore, we cannot model this scenario using a normal model.
2) To find the probability that 8 of the 10 selected students are female, we can use the binomial probability formula. The binomial probability formula calculates the probability of a specific number of successes (in this case, females) out of a fixed number of trials (total students selected) under certain conditions.
The formula to find the probability of exactly r successes in n trials with a probability p of success is: P(X = r) = nCr * p^r * (1-p)^(n-r)
In this case, we want to find the probability of exactly 8 females out of 10 students, given that the probability of selecting a female is 0.60.
P(X = 8) = 10C8 * 0.60^8 * (1-0.60)^(10-8)
Calculating the values, we have:
P(X = 8) = 45 * 0.60^8 * 0.40^2
Calculating further, we get the probability that 8 out of the 10 selected students are females.
3) To find the probability that 8 or more of the 10 students selected are female, we need to calculate the probability of having 8, 9, or 10 females.
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
Using the binomial probability formula, we can calculate each of these probabilities separately and sum them up.
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
Calculate each probability separately using the same formula as before and add them together to find the final result.