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: π‘€π‘’π‘Žπ‘› π‘œπ‘“ π‘π‘–π‘›π‘œπ‘šπ‘–π‘Žπ‘™ π‘‘π‘–π‘ π‘‘π‘Ÿπ‘–π‘π‘’π‘‘π‘–π‘œπ‘›: E(x)= Standard deviation of the distribution = 𝜎 =

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:

x=x = 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:

p=0.70p = 0.70.


Part c: Make a table that shows the probability distribution

This is a binomial distribution where:

  • n=4n = 4 (number of trials),
  • p=0.70p = 0.70 (probability of success),
  • q=1βˆ’p=0.30q = 1 - p = 0.30 (probability of failure).

The probability mass function for a binomial distribution is given by:

P(X=k)=(nk)pk(1βˆ’p)nβˆ’kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Where kk is the number of students receiving aid, n=4n = 4, and p=0.70p = 0.70.

Let's compute the probabilities for each value of kk (0, 1, 2, 3, 4):

(nk)=n!k!(nβˆ’k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Let’s calculate each probability and create a table.Here is the probability distribution for the number of students receiving financial aid (k=0,1,2,3,4k = 0, 1, 2, 3, 4):

\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)}$$ Let’s calculate both.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can easily calculate the mean and standard deviation using these formulas: - **Mean**: $$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!

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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
Οƒ = sqrt(n * p * (1 - p))

Theorems

Binomial Theorem
Law of Large Numbers

Suitable Grade Level

Grades 10-12