91Ó°ÊÓ

Consider a multiple-choice examination with 50 questions. Each question has four possible answers. Assume that a student who has done the homework and attended lectures has a \(75 \%\) probability of answering any question correctly. a. A student must answer 43 or more questions correctly to obtain a grade of A. What percentage of the students who have done their homework and attended lectures will obtain a grade of A on this multiple-choice examination? b. A student who answers 35 to 39 questions correctly will receive a grade of \(C .\) What percentage of students who have done their homework and attended lectures will obtain a grade of \(C\) on this multiple-choice examination? c. \(\quad\) A student must answer 30 or more questions correctly to pass the examination. What percentage of the students who have done their homework and attended lectures will pass the examination? d. Assume that a student has not attended class and has not done the homework for the course. Furthermore, assume that the student will simply guess at the answer to each question. What is the probability that this student will answer 30 or more questions correctly and pass the examination?

Step by step solution

01

Identify the Distribution

The scenario involves a student answering questions correctly or incorrectly, which can be modeled as a binomial distribution. The probability of success (answering a question correctly) is 0.75, and the number of questions is 50. We denote the number of correct answers as \( X \), where \( X \sim \text{Binomial}(n=50, p=0.75) \).

02

Calculate the Probability for Part (a)

We need to find \( P(X \geq 43) \). This is the probability of answering 43 or more questions correctly:\[ P(X \geq 43) = 1 - P(X \leq 42) \]Using the binomial cumulative distribution function (CDF), we find \( P(X \leq 42) \). With a binomial distribution calculator or software, compute this probability and subtract from 1.

03

Calculate the Probability for Part (b)

For a grade of C, we need the probability of answering between 35 and 39 questions correctly:\[ P(35 \leq X \leq 39) = P(X \leq 39) - P(X \leq 34) \]Use the cumulative distribution function to compute both \( P(X \leq 39) \) and \( P(X \leq 34) \), then find their difference.

04

Calculate the Probability for Part (c)

To find the probability that a student answers 30 or more questions correctly:\[ P(X \geq 30) = 1 - P(X \leq 29) \]Compute \( P(X \leq 29) \) using the binomial CDF and subtract it from 1 to get the desired probability.

05

Calculate the Probability for Part (d)

If a student guesses, the probability of answering correctly is 0.25. We are modeling this scenario as a binomial distribution, \( Y \sim \text{Binomial}(n=50, p=0.25) \).We need to find \( P(Y \geq 30) \):\[ P(Y \geq 30) = 1 - P(Y \leq 29) \]Using the binomial CDF for \( Y \), compute \( P(Y \leq 29) \) and subtract from 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability

Probability is a fundamental concept that helps us understand how likely an event is to occur. It quantifies the chances of an event happening on a scale from 0 to 1, where 0 indicates impossibility, and 1 means certainty. In a multiple-choice examination with four possible answers per question, if a student knows the material, each question has a 75% probability of being answered correctly. This 75% is represented as a probability of 0.75.

To calculate the probability of a student answering a specific number of questions correctly, we use the principles of binomial distribution. A binomial distribution is suitable here because each question can be seen as an independent trial with a fixed probability of success. For each question, there are only two possible outcomes: the student answers it correctly (success) or incorrectly (failure).

• Probability of correct answer: 0.75
• Probability of incorrect answer: 0.25
• Total number of questions (trials): 50

These parameters allow us to model the number of correct answers a student might obtain using the binomial distribution.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) is a powerful tool in statistics, particularly when dealing with binomial distributions. It represents the probability that a random variable takes a value less than or equal to a specific number. In simpler terms, it accumulates probabilities up to a certain point.

For example, in the context of the multiple-choice examination, if we want to determine the probability of a student answering at most 42 questions correctly, we use the CDF to find \( P(X \leq 42) \). This function helps us because binomial probabilities can be arduous to compute manually when dealing with exact numbers, especially in cases with a large number of trials like 50.

• The formula for a binomial CDF at a point \( k \) is the sum of all probabilities from 0 to \( k \): \( F(k) = P(X \leq k) = \sum_{x=0}^{k} \binom{n}{x}p^x(1-p)^{n-x} \)
• In our exercise, using the CDF allows us to efficiently find probabilities like \( P(X \geq 43) \) by computing \( 1 - P(X \leq 42) \)

The CDF is indispensable when solving probability problems in tests or similar scenarios where multiple outcomes must be accounted for.

Multiple-Choice Examination

A multiple-choice examination is a test format where each question has several possible answers, but only one is correct. It is a common way to assess a student's understanding and retention of the course material. In the scenario given, each question has four choices, and we're particularly interested in how well students perform given certain conditions.

For students who have diligently studied, there is a high chance of answering correctly, highlighting the importance of preparation. A well-prepared student has a 75% chance of success per question. This emphasizes the value of attending lectures and completing homework to boost one's probability of achieving a better grade.

• Grade A: Answer 43 or more questions correctly
• Grade C: Answer between 35 and 39 questions correctly
• Passing: Answer 30 or more questions correctly

Furthermore, if a student guesses answers to all questions, the probability of answering correctly drops to 25%, substantially impacting their chances of passing. This exercise effectively demonstrates how preparation, or lack thereof, influences exam outcomes, as well as provides an environment to apply probability theory practically.