Math Problem Statement

The scatterplot shows the median starting salaries and the median​ mid-career salaries for graduates at a selection of colleges. Complete parts​ (a) through​ (e) below.

​Mid-Careerequals=negative 8075−8075plus+1.9881.988

Start Med

304050607080255075100125150175Start Median Salary (thousands of $)Mid-Career Salary (thousands of $)

A graph has a horizontal axis labeled "Start Median Salary (thousands of dollars)" from 30 to 80 in increments of 10 and a vertical axis labeled "Mid-Career Salary (thousands of dollars)" from 25 to 175 in increments of 25. A line ascends from left to right and forty points are scattered around the line as it ascends, with approximately half of the points above the line and half below the line. The line travels through the following points: (30, 52), (40, 71), (50, 91), (60, 111), (70, 131) and (80, 151). All points are approximate.

Question content area bottom

Part 1

a. As the data are​ graphed, which is the independent and which the dependent​ variable?

The independent variable is

median starting salary

and the dependent variable is

median mid-career salary.

Part 2

b. Why do you suppose median salary at a school is used instead of the​ mean?

A.

Salary distributions are usually​ bimodal, making the median a more meaningful measure of center.

B.

Salary distributions are usually​ bell-shaped, making the median a more meaningful measure of center.

C.

It is easier to calculate the median for a salary distribution instead of the mean.

D.

Salary distributions are usually​ skewed, making the median a more meaningful measure of center.

Your answer is correct.

Part 3

c. Using the​ graph, estimate the median​ mid-career salary for a median starting salary of

$ 40 comma 000$40,000.

The median​ mid-career salary for a median starting salary of

$ 40 comma 000$40,000

is about

​$71,00071,000.

​(Round to the nearest thousand as​ needed.)

Part 4

d. Use the equation to predict the median​ mid-career salary for a median starting salary of

$ 40 comma 000$40,000.

The median​ mid-career salary for a median starting salary of

$ 40 comma 000$40,000

is about

​$enter your response here.

​(Round to the nearest dollar as​ needed.)

Solution

We are given the equation for predicting the median mid-career salary:

Mid-Career Salary=8075+1.988×Start Median Salary\text{Mid-Career Salary} = -8075 + 1.988 \times \text{Start Median Salary}

Step 1: Substitute 40,00040,000 for the Start Median Salary

Mid-Career Salary=8075+1.988×40,000\text{Mid-Career Salary} = -8075 + 1.988 \times 40,000

Step 2: Compute the value

Mid-Career Salary=8075+79,520\text{Mid-Career Salary} = -8075 + 79,520

=71,445= 71,445

Step 3: Round to the nearest dollar

71,445\mathbf{71,445}

Thus, the predicted median mid-career salary for a median starting salary of $40,000$ is $71,445$.

Would you like a breakdown of any step?

Related Questions:

  1. How do you determine if a variable is independent or dependent?
  2. Why is the median more robust against skewed distributions than the mean?
  3. How can we check if the regression equation is a good fit for the data?
  4. What is the significance of the slope in this regression equation?
  5. How would an outlier affect the accuracy of this prediction?

Tip: Always compare predicted values with actual values from the graph to assess the reliability of your regression model.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Scatterplots
Median Calculation
Prediction with Equations

Formulas

Mid-Career Salary = -8075 + 1.988 * Start Median Salary

Theorems

Regression Line Interpretation
Prediction using Linear Equations

Suitable Grade Level

Grades 9-12