Question

Psychologists have developed mathematical models to predict performance \(P\) (the percent of correct responses) as a function of \(n\), the number of times a task is performed. One such model is \(P=\frac{0.5+0.9(n-1)}{1+0.9(n-1)}, \quad 0

Short answer

The results of the performance calculations reveal an increasing trend according to the model, suggesting that the more a task is performed, the higher the percentage of correct responses. The limit calculation reveals that the model predicts a 100% performance rate as the task is performed an infinite number of times. The graph visually demonstrates a learning curve that substantiates these calculations.

Step-by-step solution

Compute Performance

Using the given formula \(P=\frac{0.5+0.9(n-1)}{1+0.9(n-1)}\), substitute \(n = 1,2,3,...,10\) into the formula to compute the performance value \(P\). Fill in the values in the table.

Calculate the Limit

Find the limit as \(n\) approaches infinity by substituting \(n = \infty\) into the equation. The computation simplifies to \( \lim_{n \to \infty} (\frac{0.5+0.9(n-1)}{1+0.9(n-1)}) \). By simplifying, you get 1, meaning that as the number of repeats of the task goes to infinity, the performance approaches 100%.

Graphing and Interpretation

Draw the graph using a graphing tool. The x-axis should represent \(n\) (the number of times a task is performed) and the y-axis should represent \(P\) (the percent of correct responses). The graph starts at 0.5 when \(n = 1\) and approaches 1 as \(n\) increases, which represents that the performance is improving with more repetitions of the task. This visual representation confirms the result of the limit.

Key concepts

Performance Prediction

In psychology, mathematical models are used to predict human behavior and performance. These models help us understand how performance changes with practice. For example, the given model \( P=\frac{0.5+0.9(n-1)}{1+0.9(n-1)} \) describes how a person's performance improves as they repeat a task. Here, \( n \) is the number of times the task is performed, and \( P \) is the predicted performance expressed as a percentage of correct responses.

• At \( n = 1 \), the formula computes an initial performance level.
• As \( n \) increases, the numerator (number one) and the denominator (also number one) in the formula nearly cancel each other out, improving performance.
• This type of formula highlights how consistent practice leads to improved accuracy in performing tasks.

Limit Calculation

Limit calculation is a key concept in determining the ultimate behavior of a model as certain variables increase or decrease without bound. In this exercise, we calculated the limit of the performance function \( P \) as the number of task repetitions \( n \) approaches infinity. We used the formula:\[\lim_{n \to \infty} \left( \frac{0.5+0.9(n-1)}{1+0.9(n-1)} \right)\]By simplifying the equation, we find that this limit equals 1. This result implies that with infinite practice, the performance \( P \) approaches 100%, signaling perfect performance.

• The mathematical simplification shows how repetitive practice leads to complete mastery in theoretical scenarios.
• Understanding limits helps psychologists and researchers predict long-term behaviors and outcomes.

Learning Curve

A learning curve is a graphical representation of how quickly and effectively a learner acquires new skills over time. The model in the exercise describes a learning curve that starts at a certain level and increases with practice. This curve helps us visualize performance trends and improvements:

• The initial steepness of the curve indicates rapid performance gains right after starting.
• As the curve flattens, improvements become less pronounced, indicating a move towards mastery.
• Learning curves show that most learning happens at early stages, tapering as expertise develops.

This concept is vital in educational psychology for creating effective training programs, understanding student progress, and making informed decisions about teaching methods.

Graph Interpretation

Graph interpretation involves analyzing a visual representation of data or formulas to extract meaning and insights. Using a graphing utility to plot the learning curve described by the model:

• The x-axis represents \( n \), the number of times the task is performed, while the y-axis shows \( P \), the predicted performance.
• Initially, the graph shows a steep increase in performance with further repetitions.
• Eventually, the graph levels off, reflecting that increasing practice leads to diminishing performance improvement.
• Such insights help visualize the mathematical findings, reinforcing the calculated limits and overall behavior of the learning process.

Graph interpretation is crucial for psychologists and educators to assess and communicate findings, identify trends, and refine educational strategies.