Quiz 35-Linear Regression and r-squared

Suppose the slope of the regression line is 1.055. Interpret this slope in context.

On average, if the size of a house increases by 1 square foot, we would expect the price of the house to increase by 1.055 thousands of dollars. The general interpretation of slope is: for everyone one unit increase in x, we expect y to increase/decrease by b units

The following situation applies to Questions 4-5: The distance (in feet) a cheetah runs in small bursts (distances from 1,350 to 1,600 feet) can be used to model the amount of time it takes to accelerate to its top speed (in seconds), usually about 65 mph. Use the regression output below to answer the following two questions. What is the slope of this linear regression equation? **see picture

0.027 To find the value of the slope refer to the table in the output. The value of the slope is found when looking at the cell that is the estimate of the slope.

Which of the following makes NO distinction between an explanatory variable X and a response variable Y (i.e., you can interchange the roles of X and Y and get the same result)?

Correlation Only correlation makes no distinction between the X and Y variables. With regression the line would be flipped.

The mean height of American women in their twenties is about 64 inches, and the standard deviation is about 2.7 inches. The mean height of men the same age is about 69.3 inches, with standard deviation about 2.8 inches. If the correlation between the heights of husbands and wives is about r = 0.5, what is the slope of the regression line used to predict the husband's height (Y) from the wife's height (X) in young couples? Note: b = r(sy / sx)

0.5185 For this problem we are asked to find the slope given r, and the standard deviations of our two variables. sy is going to be the standard deviation of the husband's height and sx is going to be the standard deviation of the wife's height. (The scenario told us which variables were the X and Y in the last sentance). Thus we get the equation b=.5(2.8/2.7) which comes out to be 0.5185.

Researchers collected data on the number of breeding pairs of Scarlet Macaw in an isolated area of an Amazon rainforest in each of 8 years (X) and the percentage of males who returned the next year (Y). The data show that the percentage returning is lower after successful breeding seasons and that the relationship is roughly linear. The following shows a StatCrunch regression output for these data. What percentage of the variation in the percent of returning males can be explained by the number of breeding pairs? Simple linear regression results: Dependent Variable: percent.returned Independent Variable: breeding.pairs percent.returned = 136.682 - 3.218 breeding.pairs Sample size: 8 R (correlation coefficient) = -0.8329 R-sq = 0.6937 Estimate of error standard deviation: 9.460 Parameter estimates: **see picture

69% The general interpretation of R-squared is "the percent of variating in y that is explained by the variation in x." To find the value of R-squared in the output, refer to the row that is labeled as "R-sq".

If we know the value of b, the slope of the regression line, we can accurately guess the value for the correlation coefficient without looking at the scatterplot.

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For a biology project, you measure the weight, in grams, and the tail length, in millimeters (mm), of a group of mice. The equation of the least-squares line for predicting tail length from weight is predicted tail length = 20 + 3*weight Suppose a mouse weighing 20 grams has a 78 mm tail. What is the residual for this mouse?

-2 mm A residual is how much our predicted point differs from our observed point, given a value for x. In this case we observed a mouse with a weight of 20 grams (our x value) and a tail length of 78 mm. To find our predicted point we need to plug 20 into our least squares regression line: predicted tail length=20+3*20=80. To find the residual we calculate (observed value-predicted value)=78-80= -2.

What is the value of the correlation coefficient r?

0.8219 Referring to the output look, at the row labeled "R (correlation coefficient)". This is the value of r.

The following situation applies to Questions 1-3: The size of a house (in square feet) can be used to model its selling price (in 1,000 dollars). What is the equation of the regression line from the output provided below? **see picture

Selling Price = -39.81 + 0.099*Size The general form of the regression line is y=a+bx. y represents the dependent variable which in this scenario is Price. x represents the independent variable which is Size. a represents the y-intercept. In the output table this is found at the estimate of the intercept. b represents the slope. In the output this is found at the estimate of the slope.

Suppose the y-intercept of the regression equation is -22.000. Interpret this y-intercept in context.

When the size of a house is 0 square feet, we would expect this house to sell for -22.000 thousands of dollars. The general interpretation of y-intercept is: If the average x were 0, then we would expect y to be "a".