5.7: Corequisite- Multivariable Models
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- 148626
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By the end of this lesson, you should understand that
- multivariable models have one output variable and more than one input variable.
- formulas using multiple variables are often used in real-world scenarios.
- a weighted average is a multivariable model which gives a better “average” for some sets of data.
- many problems are just variations of a weighted average problem, such as calculating grades and GPAs.
By the end of this lesson, you should be able to
- calculate the mean of a set of numbers.
- interpret and use formulas with multiple variables.
- calculate the weighted average of a set of numbers in context.
- construct an expression for a weighted average problem.
INTRODUCTION: REVIEW OF MEAN, MEDIAN, AND MODE
The terms mean, median, and mode are three different kinds of averages.
(1) (a) Match the term to the correct definition.
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(b) Imagine a place that has very different daily temperatures. The table below shows the temperatures for one week. What is the mean? What is the median? What is the mode?
Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
62° | 90° | 106° | 10° | 50° | 0° | 50° |
PROBLEM SITUATION 1: REVISITING COLLEGE RANKING
The Aspen Institute awards the Aspen Prize every other year. The award “honors those institutions that strive for and achieve exceptional levels of success for all students, while they are in college and after they graduate.” To determine the 150 finalists for the award, the Aspen Institute collects information from 1000 community colleges related to the following variables (remember that we call them variables because their values may vary between colleges):
1. Performance – The performance score measures student success in persistence, degrees awarded, completion, and transfer;
2. Change – The change score measures consistent improvement in these areas over time; and
3. Minority Achievement and Family Income – The minority achievement score and family income score measure equitable outcomes for students of all racial/ethnic and socioeconomic backgrounds.
The Aspen Institute used the following mathematical equation to calculate a score for each school. In this case, the model is represented by this equation:
\[R = 0.33*P + 0.33*C + 0.27*M + 0.07*F \nonumber\]
where:
R = Overall ranking score (higher scores are best)
P = Performance score
C = Change score
M = Minority score
F = Family income score
(2) Mathematical models that use more than one input variable to produce an output variable are called multivariable models. There are many types of multivariable models, but first we will concentrate on models which are similar to the linear models we studied previously, except that they have multiple input variables. Could the overall ranking score formula above be considered a multivariable model? Why or why not?
(3) Weighted averages are averages that give each number in a set more (or less) influence, or weight, based on what the number represents. This is different from a standard averaging calculation in which you add a set of numbers and divide the total by the count of the numbers in the set. This standard type of averaging assigns the same weight to each of the numbers (each with a weight of 1, where it is counted just one time). Weighted-averaging calculations try to give a more realistic description of some sets by giving more “valuable” numbers greater weight and therefore a greater impact on the result. Does the overall ranking score formula above represent a weighted average? Why or why not?
(4) Consider again the overall ranking score formula above.
(a) Which categories are the most significant?
(b) Which category contributes the least to the overall ranking score?
(c) If this is an “average”, what number are we dividing by?
PROBLEM SITUATION 2: CALCULATING GRADE POINT AVERAGE (GPA)
A Grade Point Average (GPA) is another kind of weighted average. Letter grades are assigned a number value in order to calculate a numerical average. This table shows typical point values:
A | A− | B+ | B | B− | C+ | C | C− | D+ | D | D− | F |
4.0 | 3.7 | 3.3 | 3.0 | 2.7 | 2.3 | 2.0 | 1.7 | 1.3 | 1.0 | 0.7 | 0 |
(5) When calculating GPA in high school, the mean is often used. With reference to both tables, calculate a student’s GPA for the following four high school classes.
Class | Grade |
English | A |
Math | B+ |
Spanish | A- |
Biology | B |
(6) When calculating a college GPA, a weighted average is used. This means some grades have a higher impact than others because the weights of the grades for each class are determined by the number of credit hours in the class. Calculate the GPA for the following four college classes. Round to the nearest hundredth.
Class | Credits | Grade |
Spanish | 5 | A- |
Physics | 4 | B |
History | 3 | C |
English | 3 | B+ |
(7) Suppose Hamm was taking the four college classes above and also got an A in her zero-credit math class. How would the math class affect her GPA?
Class | Credits | Grade |
Spanish | 5 | A- |
Physics | 4 | B |
History | 3 | C |
English | 3 | B+ |
Math | 0 | A |
(8) Suppose Messi was taking the same five college classes as Hamm and also got an F in his one-credit art class. How would the art class affect his GPA?
Class | Credits | Grade |
Spanish | 5 | A- |
Physics | 4 | B |
History | 3 | C |
English | 3 | B+ |
Math | 0 | A |
Art | 1 | F |