A regression equation for this line is
$y=-0.42 x+19.26$
What do the slope and $y$-intercept indicate?
a. The slope indicates that the percent in 1985 should have been about $19 \%$. The $y$-intercept indicates that the percent is falling at about $0.4 \%$ each year.
b. The slope indicates that the percent is falling at about $0.4 \%$ each year. The $y$-intercept indicates the percent in 1985 should have been about $19 \%$.
c. The slope indicates tha the percent has remained the same for each year. The $y$-intercept indicates the percent in 1985 should have been $22 \%$.
d. There is not enough information to determine what the slope and $y$-intercept indicate.
Answer (1)
The slope of the regression equation y = − 0.42 x + 19.26 is − 0.42 , indicating the percent is falling at about 0.4% each year.
The y -intercept is 19.26 , indicating the percent in 1985 should have been about 19% .
Therefore, the slope indicates a decrease of 0.4% per year, and the y -intercept represents the initial percentage in 1985.
The correct interpretation is that the slope indicates a decrease of 0.4% per year, and the y -intercept indicates the percent in 1985 was about 19% . b
Explanation
Analyze the equation The given regression equation is y = − 0.42 x + 19.26 . In this equation, the slope is − 0.42 and the y -intercept is 19.26 . The slope represents the rate of change of y with respect to x . Since the slope is negative, it means that as x increases, y decreases. In this context, y represents the percentage, and x represents the year. Therefore, the slope indicates that the percentage is falling by 0.42 each year. The y -intercept represents the value of y when x = 0 . If we assume that x = 0 corresponds to the year 1985, then the y -intercept indicates that the percentage in 1985 was 19.26 , which is approximately 19% .
Compare with the options Based on the analysis, the slope indicates that the percent is falling at about 0.42% each year, and the y -intercept indicates that the percent in 1985 should have been about 19.26% . Comparing this with the given options, option b is the closest to our interpretation.
Final Answer Therefore, the correct answer is: b. The slope indicates that the percent is falling at about 0.4% each year. The y -intercept indicates the percent in 1985 should have been about 19% .
Examples
Regression equations are used in various real-life scenarios. For example, a store owner might use a regression equation to predict sales based on advertising spending. If the equation is y = 2 x + 10 , where y is sales and x is advertising spending, the slope of 2 indicates that for every dollar spent on advertising, sales increase by $2. The y-intercept of 10 indicates that even with no advertising spending, the store will still make $10 in sales. This helps the store owner make informed decisions about their advertising budget.
