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.

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(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.

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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).