We continue with our historical comparisons using today’s...

We continue with our historical comparisons using today’s dollars. (Bar graph can't copy) Using today's dollars, the data in the bar graph can be described by the mathematical model $$ H=0.5 n+50 $$ where $H$ is the average nightly hotel room rate $n$ years after 1903 a. Does the formula underestimate or overestimate the average nightly hotel room rate in $2009 ?$ By how much? b. If trends shown by the formula continue, when will the average nightly hotel room rate cost $\$ 110 ?$

We continue with our historical comparisons using today’s dollars.
(Bar graph can't copy)
Using today's dollars, the data in the bar graph can be described by the mathematical model
$$
H=0.5 n+50
$$
where $H$ is the average nightly hotel room rate $n$ years after 1903
a. Does the formula underestimate or overestimate the average nightly hotel room rate in $2009 ?$ By how much?
b. If trends shown by the formula continue, when will the average nightly hotel room rate cost $\$ 110 ?$

Key Concepts

Linear Function

A linear function is a mathematical model that describes a relationship between two variables with a constant rate of change, represented by an equation in the form y = mx + b. This concept is fundamental in algebra and allows one to predict or estimate values by knowing the slope and intercept, which reflect consistent behavior over the range of the independent variable.

Slope (Rate of Change)

The slope of a linear function is a measure of how much the dependent variable changes for each unit increase in the independent variable. It represents the rate of change and is crucial in understanding the dynamics of the relationship described by the model.

Y-Intercept

The y-intercept is the value of the dependent variable when the independent variable is zero. In the context of a linear equation, it provides a starting point or initial value from which changes are measured, anchoring the model to a specific baseline.

Predictive Modeling with Linear Functions

Linear functions are often used for predictive modeling because they provide a straightforward way to forecast future values based on trends observed in past data. By substituting specific values into the equation, one can estimate outcomes and analyze when certain thresholds will be reached.

Solving Linear Equations

Solving a linear equation involves isolating the variable of interest by employing algebraic methods such as addition, subtraction, multiplication, or division. This technique is essential when determining the specific input that yields a given output in the model.

Model Estimation and Error Analysis

Estimation and error analysis in the context of linear models involve comparing predicted values with actual or estimated observed values to determine whether a model overestimates or underestimates an outcome. This process is critical for assessing the accuracy and reliability of the model.

Transcript

00:01 So using this bar graph, this bar graph shows us the median weekly earnings for women with some college degree or an associate's degree.

00:11 So in 1980, it was $231, they earned $231 a week, and in 2013 they earned $657 a week.

00:20 And we can use this equation then, also which represents the bar graphs in order to find the weekly earnings, because w represents the median weekly earnings for these women, and n is the number of years after 9%.

00:31 1980, and that's really key.

00:34 So the first question is asking us, there's a formula underestimate or overestimate the median weekly earnings in 2013.

00:42 So it's really hard to solve an equation when there are two variables.

00:47 So let's just get rid of one of the variables.

00:49 So n is a number of years after 1980.

00:52 And we're trying to solve for the weekly earnings in 2013.

00:55 So we know that 2013 is 33 years after 1980.

00:59 And we know this because we can do 2013.

01:04 Minus 1980.

01:06 And if we do that in our calculator, we get 23.

01:09 So n equals 23.

01:12 So now we can just solve this.