Questions: The least-squares regression equation is ŷ=692.2x+14,313 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 0.7867. Complete parts (a) through (d). (a) Predict the median income of a region in which 20% of adults 25 years and older have at least a bachelor's degree. (Round to the nearest dollar as needed.) This is than expected because the expected income is (Round to the nearest dollar as needed.) (c) Interpret the slope. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal. Do not round.) A. For a median income of 0, the percent of adults with a bachelor's degree is %. B. For every percent increase in adults having at least a bachelor's degree, the median income increases by , on average. C. For every dollar increase in median income, the percent of adults having at least a bachelor's degree is %, on average. D. For 0% of adults having a bachelor's degree, the median income is predicted to be . (d) Explain why it does not make sense to interpret the y-intercept Choose the correct answer below. A. It does not make sense to interpret the y-intercept because a y-value of 0 does not make sense. B. It does not make sense to interpret the y-intercept because an x-value of 0 does not make sense. C. It does not make sense to interpret the y-intercept because an x-value of 0 is outside the scope of the model. D. It does not make sense to interpret the y-intercept because a y-value of 0 is outside the scope of the model.

The least-squares regression equation is ŷ=692.2x+14,313 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 0.7867. Complete parts (a) through (d). (a) Predict the median income of a region in which 20% of adults 25 years and older have at least a bachelor's degree. (Round to the nearest dollar as needed.) This is than expected because the expected income is (Round to the nearest dollar as needed.) (c) Interpret the slope. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal. Do not round.) A. For a median income of 0, the percent of adults with a bachelor's degree is %. B. For every percent increase in adults having at least a bachelor's degree, the median income increases by , on average. C. For every dollar increase in median income, the percent of adults having at least a bachelor's degree is %, on average. D. For 0% of adults having a bachelor's degree, the median income is predicted to be . (d) Explain why it does not make sense to interpret the y-intercept Choose the correct answer below. A. It does not make sense to interpret the y-intercept because a y-value of 0 does not make sense. B. It does not make sense to interpret the y-intercept because an x-value of 0 does not make sense. C. It does not make sense to interpret the y-intercept because an x-value of 0 is outside the scope of the model. D. It does not make sense to interpret the y-intercept because a y-value of 0 is outside the scope of the model.
Transcript text: The least-squares regression equation is $\hat{y}=692.2 x+14,313$ 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 0.7867 . Complete parts (a) through (d). (a) Predict the median income of a region in which 20\% of adults 25 years and older have at least a bachelor's degree. $\$$ $\square$ (Round to the nearest dollar as needed.) This is $\square$ than expected because the expected income is \$ $\square$ (Round to the nearest dollar as needed.) (c) Interpret the slope. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal. Do not round.) A. For a median income of $\$ 0$, the percent of adults with a bachelor's degree is $\square$ \%. B. For every percent increase in adults having at least a bachelor's degree, the median income increases by $\$$ $\square$ , on average. C. For every dollar increase in median income, the percent of adults having at least a bachelor's degree is $\square$ \%, on average. D. For $0 \%$ of adults having a bachelor's degree, the median income is predicted to be $\$$ $\square$ . (d) Explain why it does not make sense to interpret the $y$-intercept Choose the correct answer below. A. It does not make sense to interpret the $y$-intercept because a $y$-value of 0 does not make sense. B. It does not make sense to interpret the $y$-intercept because an $x$-value of 0 does not make sense. C. It does not make sense to interpret the $y$-intercept because an $x$-value of 0 is outside the scope of the model. D. It does not make sense to interpret the $y$-intercept because a $y$-value of 0 is outside the scope of the model.
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Solution

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Solution Steps

Step 1: Predicting the median income

Substitute _x_ = 20 into the equation _y_ = 692.2_x_ + 14,313.

_y_ = 692.2 \* 20 + 14,313 _y_ = 13,844 + 14,313 _y_ = 28,157

Step 2: Calculating the expected income and comparison

Substitute _x_ = 28.8 into the equation _y_ = 692.2_x_ + 14,313.

_y_ = 692.2 \* 28.8 + 14,313 _y_ = 20,000.16 + 14,313 _y_ = 34,313.16

Since 37,513 > 34,313.16, the income is higher than expected.

Step 3: Interpreting the slope

The slope of the regression equation is 692.2. This means that for every one percent increase in adults having at least a bachelor's degree, the median income increases by $692.2, on average.

Final Answer:

(a) $28,157 (b) higher, $34,313 (c) B. For every percent increase in adults having at least a bachelor's degree, the median income increases by $692, on average.

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