The​ least-squares regression equation is ModifyingAbove y with caretequals737.7xplus13 comma 239 where y is the median income and x is the percentage of 25 years and older with at least a​ bachelor's degree in the region. The scatter diagram indicates a linear relation between the two variables with a correlation coefficient of 0.7679. Complete parts​ (a) through​ (d).
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Question content area top right
Part 1
15
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25
30
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20000
55000
Bachelor's %
Median Income



A scatter diagram has a horizontal axis labeled Bachelor's % from 15 to 60 in increments of 5 and a vertical axis labeled Median Income from 20000 to 55000 in increments of 5000. Points are scattered around a line that rises from left to right and has a slope of about 700.
Question content area bottom
Part 1
​(a) Predict the median income of a region in which 20​% of adults 25 years and older have at least a​ bachelor's degree.
​$
  
27,993 ​(Round to the nearest dollar as​ needed.)
Part 2
​(b) In a particular​ region, 27.9 percent of adults 25 years and older have at least a​ bachelor's degree. The median income in this region is ​$30 comma 328. Is this income higher than what you would​ expect? Why?
This is 
▼ 
lower
higher
 than expected because the expected income is ​$
  
enter your response here
​(Round to the nearest dollar as​ needed.)

Let's go through the steps for parts (a) and (b) one by one.

### (a) Predict the median income for a region where 20% of adults 25 years and older have at least a bachelor's degree.

The least-squares regression equation is given as:

\[
\hat{y} = 737.7x + 13,239
\]

Where:
- $\hat{y}$ is the predicted median income.
- $x$ is the percentage of adults with at least a bachelor's degree.

For $x = 20\%$, we can substitute into the equation:

\[
\hat{y} = 737.7(20) + 13,239
\]

Now, let's calculate:

\[
\hat{y} = 737.7 \times 20 + 13,239 = 14,754 + 13,239 = 27,993
\]

So, the predicted median income is **$27,993**.

---

### (b) In a region where 27.9% of adults have a bachelor's degree, the median income is $30,328. Is this income higher or lower than expected?

First, we will use the regression equation to predict the expected median income for $x = 27.9\%$:

\[
\hat{y} = 737.7(27.9) + 13,239
\]

Now, let's calculate:

\[
\hat{y} = 737.7 \times 27.9 + 13,239 = 20,578.83 + 13,239 = 33,818
\]

The expected income is **$33,818**.

Since the actual income is **$30,328**, which is **lower** than the expected income of $33,818,** we conclude:

- The actual income is **lower** than expected.

---

Would you like further explanation or additional details?

Here are 5 related questions:
1. How is the slope of the regression equation related to the increase in median income?
2. What does the correlation coefficient of 0.7679 indicate about the relationship between the variables?
3. How would the equation change if the percentage of bachelor's degrees had no impact on income?
4. Can you determine whether the relationship is positive or negative from the scatter plot and slope alone?
5. What assumptions are made in using linear regression for prediction?

**Tip:** In least-squares regression, the slope indicates how much the dependent variable (income) changes for each unit increase in the independent variable (education level).

The least-squares regression equation is y = 737.7x + 13,239 where y is the median income and x is the percentage of 25 years and older with at least a bachelor's degree in the region. The scatter diagram indicates a linear relation between the two variables with a correlation coefficient of 0.7679. Complete parts (a) through (d).

Explore the least-squares regression equation y = 737.7x + 13,239 to predict median income based on the percentage of adults with a bachelor's degree. Understand the linear relation between income and education with a correlation coefficient of 0.7679. Perfect for high school students studying regression analysis.

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