APPC U2.6 Competing Function Model Validation

A company records its sales over four consecutive years as follows: - Year 1: $50,000 - Year 2: $60,000 - Year 3: $72,000 - Year 4: $86,400 Which type of function (linear, quadratic, or exponential) best models the company's sales growth?

• Linear

• Quadratic

• Exponential

• None of the above

Answer explanation

The sales figures increase by a percentage each year: Year 1 to Year 2 is 20%, Year 2 to Year 3 is 20%, and Year 3 to Year 4 is 20%. This consistent percentage growth indicates an exponential function best models the sales growth.

Given the following data set representing the population of a town over five years: | Year | Population | |------|------------| | 1 | 10,000 | | 2 | 12,000 | | 3 | 14,400 | | 4 | 17,280 | | 5 | 20,736 | Which function best models the population growth?

• P(t) = 10,000 + 2,000t

• P(t) = 10,000 × 1.2^t

• P(t) = 10,000 + 2,000t + 200t^2

• P(t) = 10,000 × e^{0.2t}

Answer explanation

The population increases by a consistent percentage each year, indicating exponential growth. The function P(t) = 10,000 × 1.2^t accurately models this growth, as it reflects a 20% increase annually.

A store's monthly revenue for the past four months is as follows: - Month 1: $20,000 - Month 2: $22,000 - Month 3: $24,200 - Month 4: $26,620 Which of the following models would best represent the store's revenue over time?

• Linear

• Quadratic

• Exponential

• Cubic

Answer explanation

The revenue increases by a growing percentage each month, indicating exponential growth. The pattern shows that the revenue is not just increasing linearly, but rather at an accelerating rate, which aligns with an exponential model.

A researcher collects data on the number of bacteria in a culture over time. The data suggests that the bacteria count doubles every hour. Which function best models the bacteria population B(t) after t hours?

• B(t) = B_0 + 2t

• B(t) = B_0 × 2^t

• B(t) = B_0 × t^2

• B(t) = B_0 × e^{2t}

Answer explanation

The bacteria count doubles every hour, indicating exponential growth. The function B(t) = B_0 × 2^t correctly represents this, where B_0 is the initial count and 2^t shows the doubling effect over time.

A student fits both linear and exponential models to a data set representing the growth of a plant's height over time. After analyzing the residuals, which of the following indicates that the exponential model is more appropriate?

• Residuals increase linearly with time.

• Residuals show a random scatter with no apparent pattern.

• Residuals decrease as time increases.

• Residuals follow a quadratic pattern.

Answer explanation

Residuals showing a random scatter with no apparent pattern indicates that the model fits the data well. This suggests the exponential model captures the growth trend better than a linear model.

A data set shows that as the number of hours studied increases, test scores increase at an increasing rate. Which type of model is most appropriate for this data?

• Linear

• Quadratic

• Exponential

• Logarithmic

Answer explanation

The data shows that test scores increase at an increasing rate with hours studied, indicating a non-linear relationship. An exponential model best captures this growth pattern, unlike linear or logarithmic models.

After fitting a linear and a quadratic model to a data set, a student finds that the quadratic model has a lower total error but predicts lower values in the later part of the data range. Considering the context requires not underestimating future values, which model should the student prefer?

• Linear

• Quadratic

• Both are equally suitable

• Neither model is suitable

Answer explanation

The linear model should be preferred because it does not underestimate future values, which is crucial in this context. Although the quadratic model has lower total error, its tendency to predict lower values later is a significant drawback.

A model predicting the spread of a virus shows that the number of cases is underestimated in the later stages. What type of error is the model exhibiting in this context?

• Overestimation

• Underestimation

• Random error

• No error

Answer explanation

The model is showing underestimation because it predicts fewer cases than actually occur in the later stages of the virus spread. This indicates a systematic error in the model's predictions.

A data set regarding the depreciation of electronic devices is best modeled by which type of function?

• Linear

• Quadratic

• Exponential

• Cubic

Answer explanation

The depreciation of electronic devices typically follows an exponential decay model, as their value decreases rapidly at first and then levels off over time, making 'Exponential' the best choice.

A student is comparing a linear model and an exponential model for a given data set. The exponential model consistently overestimates the actual values at all data points. Based on model validation principles, what should the student conclude?

• The exponential model is better because it overestimates.

• The linear model is better because it does not systematically overestimate.

• Both models are equally good.

• The student needs more data to decide.

Answer explanation

The linear model is better because it does not systematically overestimate the actual values, making it more reliable for predictions compared to the exponential model, which consistently overestimates.