Brand managers become concerned if they discover that customers are aging and gradually moving out of the​ high-spending age groups. The average buyer for a particular brand of car is . A new model draws in younger customers. If a sample of purchasers of the new model has average age ​(with standard deviation ​), is this compelling evidence that the buyers of the new model are younger than the typical buyer of the​ brand? (Assume that the data meet the sample size​ condition.) Complete parts a through c. ​(a) State the null and alternative hypotheses. Describe the parameters. Choose the correct answer below. A. H0​: ​, Ha​: ​; is the average age of purchasers of the new model. B. H0​: ​, Ha​: ​; is the average age of purchasers of the new model. Your answer is not correct. C. H0​: ​, Ha​: ​; is the average age of purchasers of the new model. This is the correct answer. D. H0​: ​, Ha​: ​; is the average age of purchasers of the brand. ​(b) Describe the Type I​ (false positive) and Type II​ (false negative) errors. Choose the correct answer below. A. Concluding that purchasers of the new model are not younger when in fact they are is a Type I error. Concluding that purchasers of the new model are younger when in fact they are not is a Type II error. B. Concluding that purchasers of the new model are younger and they are is a Type I error. Concluding that purchasers of the new model are not younger when in fact they are is a Type II error. C. Concluding that purchasers of the new model are younger when in fact they are not is a Type I error. Concluding that purchasers of the new model are not younger when in fact they are is a Type II error. D. Concluding that purchasers of the new model are younger when in fact they are not is a Type I error. Concluding that purchasers of the new model are not younger and they are not is a Type II error. ​(c) Find the test statistic. Be sure to indicate its value in your Excel file. Find the​ p-value associated with the test. Do the data supply enough evidence to reject the null hypothesis if ​0.025? ​p-value 0.007 ​(Type an integer or decimal rounded to three decimal places as​ needed.) Do the data supply enough evidence to reject the null hypothesis if ​0.025? A. H0. There is sufficient evidence that purchasers of the new model are younger than years on average. B. H0. There is sufficient evidence that purchasers of the new model are younger than years on average.

answer: learning objective(s)

·         describe the creation and use of systems of equations.

·         graph a system of linear equations on the coordinate plane and identify its solution.

introduction

sometimes graphing a single linear equation is all it takes to solve a mathematical problem. other times, one line just doesn’t do it, and a second equation is needed to model the situation. this is often the case when a problem involves two variables. solving these kinds of problems requires working with a system of equations, which is a set of two or more equations containing the same unknowns.

let’s take a look at systems of equations, and see what the graphs of individual equations within a system reveal about the mathematical relationship of the variables.

systems of equations

a system of equations contains two or more linear equations that share two or more unknowns. to find a solution for a system of equations, we must find a value (or range of values) that is true for all equations in the system.

the graphs of equations within a system can tell us how many solutions exist for that system. look at the images below. each show two lines that make up a system of equations (in the graph on the right the two lines are superimposed and look like a single line). how many points in common does each system of lines reveal?

one solution

no solutions

infinite solutions

if the graphs of the equations intersect, then there is one solution that is true for both equations.

if the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

if the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.

remember, the graph of a line represents every point that is a possible solution for the equation of that line. so when the graphs of two equations cross, the point of intersection lies on both lines, meaning that it is a possible solution for both equations. when the graphs of two equations never touch, there are no shared points and there are no possible solutions for the system. when the graphs of two equations lie on top of one another, they share all their points and every one is a possible solution.

graphing as a solution method

graphing equations in order to identify a specific point of intersection is usually not a precise way to solve systems because it is often difficult to see exactly where two lines intersect (unless you are using a computer-based graphing program that allows you to zoom in on a point). however, the graph of a system of equations can still give a good idea of what type of solution, if any, exists. let’s graph a system, and see how it works.

graph the system y = 3x and x + 2y = 4. how many solutions does this system have?

a graph of the two lines y = 3x and x + 2y = 4 shows us that the lines intersect, meaning that there is a single (x, y) value that satisfies both equations. looking at the graph does not tell us exactly where that point is, but we don’t need to know that information here, because the question only asks for the number of solutions.

so a system made of two intersecting lines has one solution. now let’s look at a different situation:

how many solutions exist for the system y -0.5x = 7 and 2y = x − 3?

plotting both equations, it looks like there is no solution—the lines are parallel. to check this finding, we can compare the slopes of the equations. to make that easy, we’ll rearrange the equations into slope-intercept form, or y = mx + b.   that gives us the equations y = 0.5x + 7 and y = 0.5x − 1.5.   yes, the slope of both lines is 0.5, which means the lines really are parallel. they never intersect, so there is no point that lies on both lines, and no solution to the system.

step-by-step explanation:

a is the correct