Math Problem Statement
college claims that 70% of students receive financial aid. Suppose that 4 students at the college are randomly selected. We are interested in the number of students in the sample who receive financial aid. a. What constitutes a success on a single trial? π₯ = ___________________________________________________________________________ b. What is the probability of success on a single trial? c. Make a table that shows the probability distribution d. Find the probability that AT MOST 2 of the students in the sample receive financial aid. e. Find the probability that AT LEAST 3 of the students in the sample receive financial aid. f. Find the mean and the standard deviation using the shortcut formulas for the binomial distribution: ππππ ππ ππππππππ πππ π‘ππππ’π‘πππ: E(x)= Standard deviation of the distribution = π =
Solution
We can approach this problem as a binomial distribution problem because we are given a probability of success (receiving financial aid) and we are interested in the number of successes (students who receive financial aid) in a fixed number of trials (4 students).
Part a: What constitutes a success on a single trial?
Success on a single trial:
- A success is defined as a student receiving financial aid.
So we can define:
a student receiving financial aid.
Part b: What is the probability of success on a single trial?
The probability of success (receiving financial aid) on a single trial is given as 70%. Therefore:
.
Part c: Make a table that shows the probability distribution
This is a binomial distribution where:
- (number of trials),
- (probability of success),
- (probability of failure).
The probability mass function for a binomial distribution is given by:
Where is the number of students receiving aid, , and .
Let's compute the probabilities for each value of (0, 1, 2, 3, 4):
Letβs calculate each probability and create a table.Here is the probability distribution for the number of students receiving financial aid ():
\hline k & P(X = k) \\ \hline 0 & 0.0081 \\ 1 & 0.0756 \\ 2 & 0.2646 \\ 3 & 0.4116 \\ 4 & 0.2401 \\ \hline \end{array}$$ --- ### Part d: Find the probability that **AT MOST 2** of the students in the sample receive financial aid. This means we need to calculate $$P(X \leq 2)$$, which is the sum of the probabilities for $$X = 0, 1, 2$$. $$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$$The probability that **at most 2** of the students receive financial aid is: $$P(X \leq 2) = 0.3483$$ --- ### Part e: Find the probability that **AT LEAST 3** of the students in the sample receive financial aid. This means we need to calculate $$P(X \geq 3)$$, which is the sum of the probabilities for $$X = 3$$ and $$X = 4$$. $$P(X \geq 3) = P(X = 3) + P(X = 4)$$The probability that **at least 3** of the students receive financial aid is: $$P(X \geq 3) = 0.6517$$ --- ### Part f: Find the mean and the standard deviation using the shortcut formulas for the binomial distribution. - **Mean of the binomial distribution**: $$E(X) = n \cdot p$$ - **Standard deviation of the distribution**: $$\sigma = \sqrt{n \cdot p \cdot (1 - p)}$$ Letβs calculate both.It seems like I canβt do more advanced data analysis right now. Please try again later. However, you can easily calculate the mean and standard deviation using these formulas: - **Mean**: $$E(X) = n \cdot p = 4 \cdot 0.70 = 2.8$$ - **Standard Deviation**: $$\sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{4 \cdot 0.70 \cdot 0.30} = \sqrt{0.84} \approx 0.9165$$ Would you like more details on any part? Here are some related questions for further understanding: 1. How do binomial distributions differ from normal distributions? 2. Can you show how to calculate the binomial coefficient? 3. What are real-world examples of binomial distributions? 4. How would the probabilities change if more students were selected? 5. How do we interpret variance in the binomial distribution? **Tip**: In binomial distributions, increasing the number of trials tends to make the distribution look more like a normal distribution!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Probability
Statistics
Formulas
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
E(X) = n * p
Ο = sqrt(n * p * (1 - p))
Theorems
Binomial Theorem
Law of Large Numbers
Suitable Grade Level
Grades 10-12
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