Question

In a learning theory project, the proportion \(P\) of correct responses after \(n\) trials can be modeled by \(P=\frac{0.83}{1+e^{-0.2 n}}\) (a) Use a graphing utility to estimate the proportion of correct responses after 10 trials. Verify your result analytically. (b) Use a graphing utility to estimate the number of trials required to have a proportion of correct responses of \(0.75 .\) (c) Does the proportion of correct responses have a limit as \(n\) increases without bound? Explain your answer.

Short answer

(a) The proportion of correct responses after 10 trials is approximately 0.43. (b) It takes approximately 17 trials to reach a proportion of correct responses of 0.75. (c) Yes. The proportion of correct responses approaches 0.83 as the number of trials goes to infinity.

Step-by-step solution

Estimation in case of 10 trials

Substitute \(n = 10\) in given model. Therefore, \(P=\frac{0.83}{1+e^{-0.2*10}}\) = 0.431. Using a graphing utility will give an approximate value of 0.43.

Trials required for 0.75 correct responses

Set \(P=\frac{0.83}{1+e^{-0.2 n}} = 0.75\) and solve for \(n\). Using a graphing utility give an approximate value \(n = 17\).

Find limit of the proportion of correctness

To check if the proportion of correctness has a limit as n increases, find limit as \(n\) goes to infinity. \( \lim_{n \to \infty} \frac{0.83}{1+e^{-0.2 n}} = 0.83\). The proportion seems to approach 0.83 as \(n\) increases, hence that's the limit.

Key concepts

Sigmoid Function in Learning

The sigmoid function is a key mathematical concept in learning theory, often employed in models that describe how effectively someone or something learns over time. The general form of the sigmoid function is given by the following equation:
\[ P = \frac{L}{1 + e^{-k(x-x_0)}} \]
Where:

• \(L\) is the curve's maximum value,
• \(e\) is the base of natural logarithms,
• \(k\) is the steepness of the curve,
• and \(x_0\) is the x-value of the sigmoid's midpoint.

In the context of the given exercise, the formula
\[ P = \frac{0.83}{1 + e^{-0.2n}} \]
models the proportion of correct responses (denoted as \(P\)) after \(n\) number of trials. The function's parameters can be interpreted as follows: the maximum value of correct responses is capped at 0.83, and the growth rate is determined by the steepness factor (-0.2). Such a function is sigmoidal because it has an 'S' shaped curve, starting near 0, increasing towards its maximum value, and then leveling off. In educational scenarios, this reflects how learners may initially struggle with new material, improve rapidly with practice and then make more incremental gains as they approach mastery.

Estimation with Graphing Utility

Graphing utilities, such as graphing calculators or specialized software, are vital tools for visualizing complex mathematical functions and making estimations. These technologies allow students and researchers to easily plot the behavior of functions, such as sigmoid curves, and make predictions based on visual representations.
For example, to estimate the proportion of correct responses after a certain number of trials as in the provided exercise, one would input the function into the utility and examine the curve at the specified value of \(n\). Thus, to verify analytically the estimated value for 10 trials, you substitute \(n = 10\) into the function, yielding an estimated proportion of correct responses, which would visually correspond to a point on the curve. This demonstrates the importance of graphing utilities in making quick, intuitive estimations that can later be verified by calculation or analytical methods.

Mathematical Limits in Learning Models

Mathematical limits play a critical role in understanding learning models, particularly in determining long-term behavior as experience or trials increase. The concept of a limit in calculus refers to the value that a function approaches as the input (in this case, the number of trials \(n\)) approaches some value. In the context of learning models - especially those involved with sigmoid functions, such as the one provided in the exercise - limits help reveal the ultimate level of proficiency a learner can attain after many repetitions or trials.
In the exercise under discussion, the limit as \(n\) approaches infinity is examined to find the asymptotic value, which represents the theoretical maximum proportion of correct responses. The equation
\[ \lim_{{n \to \infty}} \frac{0.83}{1 + e^{-0.2 n}} \]
shows that as \(n\) increases without bound, the fraction approaches \(0.83\), indicating that the learner's performance will level off at this value. Understanding limits in this context is essential as it informs educators and learners about the potential effectiveness of continued practice and its diminishing returns over time.