Question 7
A study was conducted to determine whether there were significant differences between college students admitted through special programs (such as retention incentive and guaranteed placement programs) and college students admitted through the regular admissions criteria. It was found that the graduation rate was $89.8 \%$ for the college students admitted through special programs.
If 9 of the students from the special programs are randomly selected, find the probability that eactly 6 of them graduated.
\[
\text { prob }=
\]
Answer
Final Answer: The probability that exactly 6 out of 9 students from the special programs graduated is approximately \(\boxed{0.0467}\).
Step 1 ：This problem is a binomial distribution problem. The binomial distribution model is appropriate for a statistical experiment if the following conditions are met:
Step 2 ：1. The experiment consists of n repeated trials.
Step 3 ：2. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
Step 4 ：3. The probability of success, denoted by P, is the same on every trial.
Step 5 ：4. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
Step 6 ：In this case, we have n=9 (number of trials), p=0.898 (probability of success - graduation), and we need to find the probability that x=6 (number of successes - graduates).
Step 7 ：The formula for binomial distribution is: \[P(x; n, p) = \binom{n}{x} * p^x * (1 - p)^{n - x}\] where:
Step 8 ：- P(x; n, p) is the probability of getting exactly x successes in n trials
Step 9 ：- \(\binom{n}{x}\) is the number of combinations of n items taken x at a time
Step 10 ：- p is the probability of success on any given trial
Step 11 ：- x is the number of successes
Step 12 ：- n is the number of trials
Step 13 ：Substituting the given values into the formula, we get: \[P(6; 9, 0.898) = \binom{9}{6} * 0.898^6 * (1 - 0.898)^{9 - 6}\]
Step 14 ：Calculating the above expression, we get: 0.04674528601057579
Step 15 ：Final Answer: The probability that exactly 6 out of 9 students from the special programs graduated is approximately \(\boxed{0.0467}\).
