No, it is not possible to interpret the y-intercept in this context because the y-intercept is negative, and weights cannot be negative.
When growing giant pumpkins for competitions, growers need to keep track of the weights of the pumpkins while they are growing. It is difficult to weigh a large pumpkin before it is harvested, so a method has been developed for estimating the weight. The grower measures around the pumpkin both horizontally and vertically, then adds the results. This is called the OTT (over the top) measurement and is used to predict the weight of the pumpkin. Following are the OTT measurements and actual weights of the 10 largest pumpkins entered into official competitions in a recent year.
OTT (inches) Weight (pounds)
469 2469
465 2138
456 2077
477 2157.5
490 2433.9
465 2152
462 2070.1
480 2416.5
457 2079
490 2528
The regression line is y= -3449.244+12.1022x
Is it possible to interpret the y-intercept?
[yes or no?] because the y intercept is [positive or negative] and weights cannot be [positive or negative]?
5 answers
(c) If two pumpkins differ in OTT by 7 inches, by how much would you predict their weights to differ? Round the answer to two decimal places as needed.
To find the predicted difference in weights of two pumpkins that differ in OTT by 7 inches, you can use the slope of the regression equation.
The regression line is given as:
\[ y = -3449.244 + 12.1022x \]
The slope (12.1022) indicates how much the weight (y) changes for each unit change in OTT (x).
If the OTTs of the pumpkins differ by 7 inches, the difference in weights (Δy) can be calculated as follows:
\[ \Delta y = \text{slope} \times \Delta x \] \[ \Delta y = 12.1022 \times 7 \]
Calculating this:
\[ \Delta y = 12.1022 \times 7 = 84.7154 \]
Rounding to two decimal places:
\[ \Delta y \approx 84.72 \]
So, if two pumpkins differ in OTT by 7 inches, you would predict their weights to differ by approximately 84.72 pounds.
(d) Predict the weight of a pumpkin whose OTT is 470 inches. Round the answer to two decimal places as needed.
To predict the weight of a pumpkin with an OTT of 470 inches using the regression equation provided, you can substitute \(x = 470\) into the regression line equation:
\[ y = -3449.244 + 12.1022x \]
Now, replace \(x\) with 470 inches:
\[ y = -3449.244 + 12.1022 \times 470 \]
Calculating the product:
\[ 12.1022 \times 470 = 5687.034 \]
Now, substitute back into the equation:
\[ y = -3449.244 + 5687.034 \]
Calculating the result:
\[ y = 5687.034 - 3449.244 = 2237.79 \]
Thus, the predicted weight of a pumpkin with an OTT of 470 inches is approximately 2237.79 pounds.