Question

What is the variance of the number of right answers for someone who knows \(80 \%\) of the material on which a 25 -question quiz is based? What if the quiz has 100 questions? 400 questions? How can you "correct" these variances for the fact that the "spread" in the histogram for the "number of right answers" random variable only doubled when the number of questions in a test was quadrupled?

Short answer

Variances are 4, 16, and 64 for 25, 100, and 400 questions, respectively. Corrected variance: 0.16.

Step-by-step solution

Understand the Problem Statement

The exercise asks us to find the variance of the number of correct answers given someone has knowledge of 80% of the quiz material. We will address the problem for quizzes with 25, 100, and 400 questions. Additionally, it asks us to adjust these variances considering the 'spread' when the test size changes.

Define the Random Variable

Let the random variable \( X \) represent the number of right answers. Since each question can be viewed as a Bernoulli trial with probability \( p = 0.8 \) of success (right answer), \( X \) follows a binomial distribution.

Recall Variance Formula for Binomial Distribution

The variance of a binomial distribution \( X \sim \text{Binomial}(n, p) \) is given by the formula: $$ \text{Var}(X) = n \cdot p \cdot (1-p). $$ We'll use this formula to calculate variance for different quiz sizes.

Calculate Variance for 25 Questions

Given \( n = 25 \) and \( p = 0.8 \), find the variance: \[ \text{Var}(X) = 25 \times 0.8 \times 0.2 = 4. \]

Calculate Variance for 100 Questions

With \( n = 100 \) and \( p = 0.8 \), the variance is calculated as: \[ \text{Var}(X) = 100 \times 0.8 \times 0.2 = 16. \]

Calculate Variance for 400 Questions

For \( n = 400 \) and \( p = 0.8 \), compute the variance as: \[ \text{Var}(X) = 400 \times 0.8 \times 0.2 = 64. \]

Analyze and Correct for Spread Change

The spread (standard deviation) is the square root of the variance. Observing the variance as \( n \) quadruples (from 100 to 400), the variance increases by factor 4. To adjust for spread changing linearly with \( n \), divide variance by \( n \), making it: \( \text{Var}_{\text{corrected}} = p(1-p) \).

Key concepts

binomial distribution

The concept of binomial distribution is key when analyzing probabilities that involve yes/no outcomes. This probability distribution applies to tasks where there are only two potential results - think of it as the mathematically idealized coin toss. In our exercise scenario involving multiple choice questions:

• Each question can either be answered correctly or incorrectly.
• The probability of getting a question correct is constant across all trials.

When conditions such as a fixed number of trials, independent trials, and same probability of success on each trial are met, we refer to this situation as a binomial distribution.
It is characterized by two key parameters:

• n: the number of trials (questions).
• p: the probability of success (answering correctly).

This framework allows us to calculate probabilities for different outcomes in scenarios like our quiz problem.

Bernoulli trial

A Bernoulli trial is a single experiment or observation in which there are exactly two possible outcomes - typically labeled as "success" and "failure." Each question in our quiz is effectively a Bernoulli trial because:

• It has two possible outcomes: the student gets the question right (success) or wrong (failure).
• The probability of success (getting it right) is constant for each question, i.e., 0.8 in our scenario.

These trials are independently conducted, which means the probability of success remains unchanged regardless of the outcome of other trials.
The concept is crucial because the aggregation of many Bernoulli trials leads to a binomial distribution, providing us with a powerful tool for modeling and understanding probabilities across a given set of repeated trials.

standard deviation

The standard deviation is a critical concept when discussing how data points are spread out in a distribution. It is derived as the square root of the variance, which quantifies the average squared deviation from the mean.

• In the given exercise, variance is calculated using the binomial distribution formula: \[ \text{Var}(X) = n \cdot p \cdot (1-p) \]
• The standard deviation \( \sigma \) is thereby: \[ \sigma = \sqrt{\text{Var}(X)} \]

Standard deviation gives a more intuitive measure of spread or "spread out-ness" because it expresses average deviation in the same units as the data itself.
In our quiz context, as the number of questions increases, both variance and standard deviation scale, altering the shape and spread of the overall distribution.

probability theory

Probability theory is the branch of mathematics that deals with analyzing random phenomena. It provides the foundation for understanding all the other concepts mentioned here.

• It introduces the notion of a 'random variable,' which in our case symbolizes the number of right answers on the quizzes.
• This theory helps us determine the likelihood of different outcomes happening - like a student getting a certain number of questions correct.

Probability theory provides both the language and tools for calculating the chances of events, guiding decisions under uncertainty.
It closely relates to our problem because it informs how calculations of likelihood, variance, and standard deviation should be understood and interpreted in practical scenarios, such as student performance on a test with varying numbers of questions.