2.8.2: Exercise M.8
- Page ID
- 148571
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Consumer Reports rating for cereals is created using numerous factors34 and indicates how healthy a cereal is for you. An instructor at Yale used a “Healthy Breakfast” dataset to analyze how the Consumer Reports rating for various cereals is related to the nutrition information on the cereals’ packages.35 In particular, the instructor looked at the amount of fat, sugar, and fiber in the cereal. He found that the following was the best model for how these nutritional components related to the Consumer Reports rating, where fat, fiber, and sugar are all in grams per serving:
Rating = 53.4 − (3.48 * Fat) + (2.95 * Fiber) − (1.96 * Sugar)
(1) Maria wanted to know what the Consumer Reports rating for her three favorite cereals would be, but all she had was the nutrition labels from the cereal boxes. She decided to use the equation above to calculate the rating. The information from the cereal boxes is shown in the table below.
| Cereal 1 | Cereal 2 | Cereal 3 | |
| Fat | 2 | 11 | 3 |
| Fiber | 3 | 12 | 2 |
| Sugar | 1 | 4 | 9 |
| Rating |
- Calculate the rating for each cereal and enter it into the table. Round to two decimal places.
(b) If Maria wanted to eat the most healthy cereal (the cereal with the highest rating), which cereal should she choose?
(c) Interpret the coefficient for Fiber in the model above.
In Collaboration M.8, you saw examples of weighted averages using relative weights.
(2) Calculate the weighted averages for two additional used cars using the weights given in the table below.
| New weights | Car 4 Ratings | Car 5 Ratings | ||
| Reliability | 5 | 7 | 6 | |
| Gas Mileage | 6 | 5 | 7 | |
| Interior Features/Comfort | 10 | 8 | 7 | |
| Cargo Space | 4 | 6 | 8 |
(a) Weighted average for Car 4:
(b) Weighted average for Car 5:
(3) Weighted averages can also be created using percentages as weights.
(a) In your weighted average equations above, how many total weights are used?
(b) For each category, what percent of the total weights are represented by that category’s weight? Enter your answers in the table.
| New Weights as a Percentage of Total Weights | |
| Reliability | |
| Gas Mileage | |
| Interior Features/Comfort | |
| Cargo Space |
(c) In the tables below, multiply each category’s percentage weight by its rating, and add up the four amounts.
| Percentage Weights (written as a decimal) | Car 4 Ratings | ||||
| Reliability | x | 7 | = | ||
| Gas Mileage | x | 5 | = | ||
| Interior Features/Comfort | x | 8 | = | ||
| Cargo Space | x | 6 | = | ||
| Total | |||||
| Percentage Weights (written as a decimal) | Car 5 Ratings | ||||
| Reliability | x | 6 | = | ||
| Gas Mileage | x | 7 | = | ||
| Interior Features/Comfort | x | 7 | = | ||
| Cargo Space | x | 8 | = | ||
| Total | |||||
(d) Write an equation for the weighted average using the percentage weights. Use upper case letters to represent the categories (e.g. R, G, I, and C).
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 |
When calculating a GPA, some grades have a higher impact than others because the weights of the grades for each course are determined by the number of credit hours in the course.
(4) Jose gets the following grades in his courses his first term in college. What would his Grade Point Average (GPA) be for that term? Enter your answer rounded to two decimal places. (Hint: You will need to change each letter grade into a point value and then use the number of credits as the weight for each grade. Remember that you need to use the total number of weights [credits] in your calculation.)
| Class | Credits | Grade |
| Pre-calculus | 5 | B |
| Spanish I | 5 | A- |
| Chemistry I (with lab) | 6 | B+ |
| Archery | 1 | A |
To calculate a GPA after several terms, you can think of the current GPA as being weighted by the total number of credits taken in previous terms, since the credit-hour weights of the courses the student has taken previously have already been used in the calculation of the GPA from previous terms. For example, suppose another student currently has a GPA of 2.85 after 40 credits and now completes four new courses. Then that student’s GPA can be calculated from this information:
| Class | Credits | Grade | Point Value |
| Pre-calculus | 5 | B | 3.0 |
| Spanish I | 5 | A− | 3.7 |
| Chemistry I (with lab) | 6 | B+ | 3.3 |
| Archery | 1 | A | 4.0 |
| Credits taken in previous terms | 40 | -- | 2.85 |
| Total | 57 |
(5) Calculate this student’s GPA. Round to the nearest hundredth.
(6) Suppose Alex wants to get a C+ or better in his math class and his instructor says a C+ is 75%–77% for that course. Below are Alex’s grades and his teacher’s weightings of different assessments. Calculate the minimum grade (as a percent) Alex must earn on the final to get a 75% or higher in the class. Assume that the percent on the final exam must be a whole number. (Hint: One approach is to set up an equation using the grades and the weighted percentages, then use a variable to represent the unknown final exam grade. The weighted average should equal the minimum score Alex would need to get on the final exam in order to earn a C+ in the class.)
| Assessment | Grade | Percentage Weight |
| Homework | 76% (average) | 10% |
| Quizzes | 82% (average) | 20% (total) |
| Midterms (3) | 65%, 60%, 72% | 10% each |
| Final Exam | ??? | 40% |
(7) A college student, Andrea, wants to apply for a scholarship. The scholarship requires a minimum GPA of 2.5. Andrea has a GPA of 2.3 after completing 70 college credits. She is taking 16 credits this term. What is the minimum GPA Andrea will need this term in order to be eligible for the scholarship at the end of this term?
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