**** Suppose Barbara has a 

1515​-minute 

commute and scores 

66.766.7 

on the survey. Is Barbara more​ "well-off" than the typical individual who has a 

1515​-minute 

​commute? Select the correct choice below and fill in the answer box to complete your choice.

​(Round to one decimal place as​ needed.)

A.

​Yes, Barbara is more​ well-off because the typical individual who has a 

1515​-minute 

commute scores 

enter your response here.

B.

​No, Barbara is less​ well-off because the typical individual who has a 

1515​-minute 

commute scores 

enter your response here. Commute Time (min),x	Well-Being Index score,y

5	68.9

20	67.2

25	66.4

40	65.1

60	63.3

84	61.8

105	59.3

We are tasked with determining if Barbara is more or less "well-off" than the typical individual with a 15-minute commute, based on their survey scores.

From the data provided, we know the well-being index score \( y \) depends on the commute time \( x \). The available data points are:

\[
\begin{array}{|c|c|}
\hline
\text{Commute Time (min), } x & \text{Well-Being Index score, } y \\
\hline
5 & 68.9 \\
20 & 67.2 \\
25 & 66.4 \\
40 & 65.1 \\
60 & 63.3 \\
84 & 61.8 \\
105 & 59.3 \\
\hline
\end{array}
\]

Since the commute time of interest is 15 minutes, which is not directly listed, we will need to estimate the well-being index score for 15 minutes using the data from 5 and 20 minutes. We can do this using **linear interpolation**.

### Step 1: Linear Interpolation Formula
To find the score at 15 minutes, we use the formula for linear interpolation:

\[
y = y_1 + \frac{(x - x_1)}{(x_2 - x_1)} \times (y_2 - y_1)
\]

Where:
- \( x_1 = 5 \) (commute time closest below 15),
- \( x_2 = 20 \) (commute time closest above 15),
- \( y_1 = 68.9 \) (well-being score at 5 minutes),
- \( y_2 = 67.2 \) (well-being score at 20 minutes),
- \( x = 15 \) (commute time of interest).

### Step 2: Calculating the Estimated Well-Being Score for 15 Minutes
Let's calculate the estimated score for a 15-minute commute.The estimated well-being score for a 15-minute commute is approximately **67.8**.

### Step 3: Comparing Barbara's Score
- Barbara's score is **66.7**.
- The estimated score for a typical individual with a 15-minute commute is **67.8**.

### Conclusion
Since Barbara's score is lower than the typical individual's score, the correct answer is:

**B. No, Barbara is less well-off because the typical individual who has a 15-minute commute scores 67.8.**

---

Do you want any further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do we calculate linear interpolation for larger data sets?
2. What if we used polynomial interpolation instead of linear?
3. How does commute time generally affect well-being according to this data?
4. What happens to well-being if Barbara's commute were to increase?
5. Can you plot the well-being score as a function of commute time?

**Tip**: When working with scattered data, interpolation methods (linear or polynomial) are useful for estimating values between known data points.

Suppose Barbara has a 15-minute commute and scores 66.7 on the survey. Is Barbara more "well-off" than the typical individual who has a 15-minute commute? Select the correct choice below and fill in the answer box to complete your choice.

A. Yes, Barbara is more well-off because the typical individual who has a 15-minute commute scores [enter your response here].

B. No, Barbara is less well-off because the typical individual who has a 15-minute commute scores [enter your response here].

Commute Time (min), x | Well-Being Index score, y:
5 | 68.9
20 | 67.2
25 | 66.4
40 | 65.1
60 | 63.3
84 | 61.8
105 | 59.3

Learn how to estimate the well-being score for a 15-minute commute using linear interpolation. This example compares Barbara's well-being score with the typical score for the same commute time, offering a step-by-step solution. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation